# Geometry

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See Geometry (encyc) in the encyclopedia.

## Basics

For these pairs of points, find the midpoint, distance, slope, and equation of the line.

1. solution $\displaystyle (-2,10),(4,9)\,$ .

2. solution $\displaystyle (12,1),(4,0)\,$ .

3. solution $\displaystyle (4,13),(0,9)\,$ .

4. solution $\displaystyle (-5,15),(-5,9)\,$ .

5. solution $\displaystyle (0,9),(6,-9)\,$ .

6. solution Find the value of a if the distance between the points $\displaystyle (a,2),(3,4)\,$ is 2.

7. solution Find the relation between x and y,if the point (x,y)is equidistant from $\displaystyle (7,-6),(-3,4)\,$

8. solution Find the area of the triangle formed by $\displaystyle (5,2),(-9,-3),(-3,-5)\,$

9. solution Find the area of the triangle formed by $\displaystyle (1,3),(3,-1),(-5,-5)\,$

10. solution Find the area of the triangle formed by $\displaystyle (2,4),(2,6),(2+\sqrt{3},5)\,$

11. solution Find the value of x if the area of the triangle formed by $\displaystyle (10,2),(-3,-4),(x,1)\,$ is 5 square units

12. solution Find the centroid of a triangle whose vertices are formed by $\displaystyle (2,7),(3,-1),(-5,6)\,$

13. solution Find the point which divides the line segment joining (-1,2) and (4,-5) in the ratio 3:2

14. solution Find the point which divides the line segment joining (4,5) and (-3,4) in the ratio -6:5

15. solution Find the ratio in which the point P(2,1)divides the line segment joining the points A(1,-2) and B(4,7).

16. solution Find the ratio in which the X-axis divides the line joining the points (2,4) and (-4,3).

17. solution Show that the triangle formed by the points(4,4),(3,5) and (-1,-1) is a right-angled triangle.

18. solution Show that the points (-1,7),(3,-5),(4,-8) are collinear.

19. solution Find the value of k,if the points (k,2-2k),(-k+1,2k),(-4-k,6-2k) are collinear

20. solution Find the equation of the locus of points which are 5 units away from A(4,-3).

21. solution Find the equation of the locus of points which are equidistant from $\displaystyle A(-3,2),B(0,4)\,$

22. solution Find the equation of locus of points P such that distance of P from origin is twice the distance of P from $\displaystyle A(1,2)\,$

23. solution Given that the points $\displaystyle A(2,3),B(-3,4)\,$ are points on a triangle, find the locus of P such that the area of the triangle PAB is 8.5 square units.

24. solution Find the locus of a point P, the square of whose distance from origin is 4 times its y coordinate.

25. solution Find the locus of P if the ratio of the distances from P to $\displaystyle (5,-4),(7,6)\,$ is 2:3.

## Straight Lines-I

1.solution Find the equation of the straight line making an angle of $\displaystyle 120^\circ\,$ with the X-axis in positive direction and passing through the point $\displaystyle (0,-2)\,$

2.solution Find the equation of the straight line which makes intercepts 5 and 6 on the X and Y-axis respectively.

3.solution Find the equation of the straight line which makes intercepts whose sum is 5 and product is 6.

4.solution Find the equation of the straight line passing through the point $\displaystyle (2,3)\,$ and making intercepts whose sum is zero.

5.solution Find the slope of the straight line joining $\displaystyle (-3,8),(10,5)\,$

6.solution Find the value of x,if the slope of the line joining $\displaystyle (2,5),(x,3)\,$ is 2.

7.solution Find the equation of the straight line passing through the points $\displaystyle (1,1),(2,3)\,$

8.solution Find the equation of a straight line joining the points $\displaystyle (at_1^2,2at_1),(at_2^2,2at_2)\,$

9.solution Find the value of y, if the line joining$\displaystyle (3,y),(2,7)\,$ i sparallel to the line joining $\displaystyle (-1,4),(0,6)\,$

10.solution Find the equation of the straight line which makes $\displaystyle 135^\circ\,$ with X-axis and passing through $\displaystyle (3,-2)\,$

11.solution Find the equation of the straight line passing through $\displaystyle (-4,5)\,$ and cutting off equal intercepts on the coordinate axes.

12.solution Find the equation of the straight line passing through $\displaystyle (3,-4)\,$ and making intercepts in the ratio 2:3.

13.solution Find the equation of the straight line passing through $\displaystyle (4,-3)\,$ and perpendicular to the line joining $\displaystyle (1,1),(2,3)\,$ .

14.solution Find the equation of the straight line passing through $\displaystyle (4,-3)\,$ and parallel tothe line joining $\displaystyle (6,3),(-4,5)\,$ .

15.solution Show that the points $\displaystyle (-5,1),(5,5),(10,7)\,$ are collinear.

16.solution Show that the points $\displaystyle (a,b+c),(b,c+a),(c,a+b)\,$ are collinear.

17.solution $\displaystyle A(10,4),B(-4,9),C(-2,-1)\,$ are the vertices of a triangle.Find the equations of i)AB. ii)Median through A. iii)Altitude through B. iv)Perpendicular bisector of side AB.

18.solution Find the equation of the straight line whose distance from origin is 4units and the normal from the orgin to the straight line makes an angle of $\displaystyle 135^\circ\,$ with the X-axis in positive direction.

19.solution Show that the equation of the straight line passing through $\displaystyle (x_1,y_1)\,$ and making an angle of $\displaystyle \theta\,$ with the X-axis in positive direction is $\displaystyle \frac{x-x_1}{\cos\theta}=\frac{y-y_1}{\sin \theta}\,$ .

20.solution Find the equation of the straight line in symmetric form having slope $\displaystyle \sqrt{3}\,$ and passing through $\displaystyle (2,3)\,$ .

21.solution Find the equation of the straight line in symmetric form having slope $\displaystyle \frac{-1}{\sqrt{3}}\,$ and passing through $\displaystyle (-2,0)\,$ .

22.solution Distance of a straight line from the origin is p.The normal on the straight line from the origin makes an angle $\displaystyle \alpha\,$ with the X-axis in positive direction.Find the equations of straight lines whose values are $\displaystyle p=5,\alpha=60^\circ\,$

23.solution Write the various forms of equation of a straight line.

24.Theorm If the equations $\displaystyle a_1x+b_1y+c=0,a_2+b_2+c_2=0\,$ represent the same straight line then prove that $\displaystyle \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\,$

25.Theorm Change the equation $\displaystyle ax+by+c=0\,$ into normal form.

26.Solution Transform the equation $\displaystyle 5x-2y-7=0\,$ into i)Slope-intercept form ii).Intercept form iii).Normal form.

27.Solution Transform the equation $\displaystyle \sqrt{3}x+y+10=0\,$ into normal form.

28.Theorm The ratio in which the straight line $\displaystyle L=ax+by+c=0\,$ divides the line joining the points $\displaystyle A(x_1,y_1),B(x_2,y_2)\,$ is $\displaystyle \frac{-L_[x_1,y_1]}{L_[x_2,y_2]}\,$ .

29.Solution Find the ratio in which the straight line$\displaystyle 2x+3y-20=0\,$ divides the line joining the points $\displaystyle (2,3),(7,10)\,$

30.Solution Find the ratio in which the straight line$\displaystyle 3x-4y-7=0\,$ divides the line joining the points $\displaystyle (2,-7),(-1,3)\,$

31.Solution If $\displaystyle (2,1),(-5,7),(-5,-5)\,$ are mid points of sides of a triangle,find the equations of the sides of a triangle.

32.Solution Find the point on the straight line $\displaystyle 3x+y+4=0\,$ which is equidistant from the points $\displaystyle (-5,6),(3,2)\,$

33.Solution If the perpendicular distance of the straight line $\displaystyle \frac{x}{a}+\frac{y}{b}=1\,$ is p, prove that $\displaystyle \frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}\,$ .

34.Solution Find the equation of the straight line passing through the point of intersection of the lines $\displaystyle 5x-3y-3=0,4x+y-1=0\,$ and passing through the point $\displaystyle (0,2)\,$

35.Solution Find the equation of the altitude from A to side BC of triangle ABC formed by $\displaystyle A(1,1),B(-3,4),C(2,-5)\,$

36.Solution Find the equations of the medians of the triangle formed by $\displaystyle (2,-5),(-3,4),(0,-3)\,$

37.Solution Show that the feet of the perpendicular from $\displaystyle (-2,8)\,$ to the lines $\displaystyle x-2y+6=0,x-7y+6=0,x+3y-4=0\,$ are collinear.

38.Solution The three straight lines $\displaystyle px+qy+r=0,qx+ry+p=0,rx+py+q=0\,$ are concurrent if $\displaystyle p+q+r=0\,$

39.Solution Prove that the perpendicular bisectors of the sides of a triangle are concurrent.

40.Solution Find the point of intersection of diagonals of the quadrilateral with vertices $\displaystyle (1,2),(3,4),(2,1),(-1,-2)\,$

41.Solution Find the value of k if the lines $\displaystyle 2x+3y=6,kx-9y+4=0,5x+6y=13\,$ are concurrent.

42.Solution A variable straight line drawn through the point of intersection of the straight lines $\displaystyle \frac{x}{a}+\frac{y}{b}=1\,$ and $\displaystyle \frac{x}{b}+\frac{y}{a}=1\,$ meets the coordinate axes at A and B. Show that the locus of the midpoint of AB is $\displaystyle 2(a+b)xy=ab(x+y)\,$

43.Solution Show that the four lines $\displaystyle ax\pm by\pm c=0\,$ form a rhombus whose area is $\displaystyle \frac{2c^2}{ab}\,$ .

44.Solution Find the circumcenter of the triangle formed by the points $\displaystyle (2,1),(1,-2),(-2,3)\,$

45.Solution Two vertices of a triangle are $\displaystyle (5,-1),(-2,3)\,$ . If the orthocenter of the triangle is the origin,find the third vertex.

## Straight Lines-II

1.solution Find the equation to the pair of lines passing through the origin and perpendicular to the pair $\displaystyle ax^2+2hxy+by^2=0\,$ is $\displaystyle bx^2-2hxy+ay^2=0\,$

2.solution Find the equation to the pair of lines through the origin and forming an equilateral triangle with the line $\displaystyle x-2y-1=0\,$ .Find also the area of the triangle.

3.solution Find the condition that the lines represented by $\displaystyle ax^2+2hxy+by^2=0\,$ are such that the slope of one line is $\displaystyle \lambda\,$ times that of the other.

4.solution Find the area of the triangle formed by the lines $\displaystyle x^2+4xy+y^2=0,x+y=1\,$

5.solution Find the equation to the two lines represented by the equation $\displaystyle x^2-2xy\csc\theta+y^2=0\,$

6.solution Find the centroid of the triangle formed by the lines $\displaystyle x^2-5xy+4y^2=0\,$ and $\displaystyle x+2y-6=0\,$

7.solution Show that if one of the lines given by $\displaystyle a_1 x^2+2h_1 xy+b_1 y^2=0\,$ coincides with one of the lines of $\displaystyle a_2 x^2+2h_2 xy+b_2 y^2=0\,$ then $\displaystyle (a_1 b_2-a_2 b_1)^2=4(a_2h_1-a_1h_2)(b_1 h_2-b_2 h_1)\,$

8.solution Show that the lines $\displaystyle x^2+16xy-11y^2=0\,$ form an equilateral triangle with the line $\displaystyle 2x+y+1=0\,$ and find its area.

9.solution The distance of a point $\displaystyle P(h,k)\,$ from a pair of lines passing thro'the origin is d units.Show that the equation of the pair of lines is $\displaystyle (xk-hy)^2=d^2(x^2+y^2)\,$

10.solution If $\displaystyle ax^2+2hxy+by^2=0\,$ be two sides of a parallelogram and $\displaystyle px+qy=1\,$ is one diagonal,prove that the other diagonal is $\displaystyle y(bp-hq)=x(aq-hp)\,$

11.solution Find the equation to the pair of angle bisectors of the pair of lines $\displaystyle (ax+by)^2=3(bx-ay)^2\,$

12.solution If the pair of line $\displaystyle ax^2+2pxy-ay^2=0,bx^2+2qxy-by^2=0\,$ are such that each pair bisects the angle between the other pair,then show that $\displaystyle ab+pq=0\,$

13.solution Prove that one of the lines $\displaystyle ax^2+2hxy+by^2=0\,$ will bisect the angle between the coordinate axes if $\displaystyle (a+b)^2=4h^2\,$

14.solution Prove that the pair of lines $\displaystyle (a-\lambda)x^2+2hxy+(b-\lambda)y^2=0\,$ is equally inclined with the pair $\displaystyle ax^2+2hxy+by^2=0\,$

15.solution Find the bisecting line of the acute angle between the lines$\displaystyle 3x-4y+2=0,-12x+5y-2=0\,$

16.solution Find the value of k for which the equation $\displaystyle 12x^2-10xy+2y^2+14x-5y+k=0\,$ represents two straight lines. Find their point of intersection.

17.solution Find the value of k for which the equation $\displaystyle 2x^2+2xy-y^2+kx+6y-9=0\,$ represents two straight lines. Find their point of intersection.

18.solution Find the equation to the pair of bisectors of angles between $\displaystyle 3x^2+4xy-4y^2-11x+2y+6=0\,$

19.solution Find the equation of the lines which pass through the point of intersection of the pair of lines $\displaystyle x^2-5xy+4y^2+x+2y-2=0\,$ and are at right angles to them.

20.solution If $\displaystyle 2x^2+3xy+py^2-5x+10y+q=0\,$ represents a pair of perpendicular lines,find p,q.Find their point of intersection.

21.solution If $\displaystyle ax^2+2hxy+by^2+2gx+2fy+c=0\,$ represents a pair of lines then show that the square of the distance from the origin to their point of intersection is $\displaystyle \frac{c(a+b)-f^2-g^2}{ab-h^2}\,$

22.solution Find k if the equation $\displaystyle 4x^2+2lxy+y^2+6x+ky=10\,$ represent a pair of parallel lines.Also find the distance between them.

23.solution The equation $\displaystyle ax^2+2hxy+by^2+2gx+2fy+c=0\,$ represents a pair of parallel lines. Prove that the equation of the line midway between the two parallel lines is $\displaystyle hx+by+f=0\,$

24.solution Show that the pair of lines $\displaystyle 6x^2+5xy-4y^2+7x+13y-3=0\,$ form a parallelogram with the pair of lines $\displaystyle 6x^2+5xy-4y^2=0\,$ .Find its area.

25.solution Show that the two pairs of lines $\displaystyle 12x^2+7xy-12y^2=0,12x^2+7xy-12y^2-x+7y-1=0\,$ form a square.

26.solution Show that the lines joining the origin to the points of intersection of two curves $\displaystyle ax^2+2hxy+by^2+2gx=0,a_1 x^2+2h_1 xy+by_1^{2}+2g_1 x=0\,$ will be at right angles to one another if $\displaystyle g(a_1+b_1)=g_1(a+b)\,$

27.solution If the chord $\displaystyle x+y=b\,$ of the curve $\displaystyle x^2+y^2-2ax-4a^2=0\,$ subtends a right angle at the origin ,prove that $\displaystyle b(b-a)=4a^2\,$

## Circles

1. i). The equation of a circle whose centre is (a,b) and radius r is

$\displaystyle (x-a)^2+(y-b)^2=r^2\,$

ii). The equation of a circle is $\displaystyle x^2+y^2+2gx+2fy+c=0\,$

radius is $\displaystyle \sqrt{g^2+f^2-c}\,$ centre is (-g,-f)

iii). Equation of the circle described on the line segment AB where A=(x1,y1),B=(x2,y2) is

$\displaystyle (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0\,$

2.solution Find the equation to the circle of radius 3 and centre (3,-2).

3.solution Find the equation of the circle which passes through (-7,1) and has centre (-4,-3).

4.solution Find the centre and radius of the circle $\displaystyle x^2+y^2+6x+8y-96=0\,$

5.solution Find the centre and radius of the circle $\displaystyle 5x^2+5y^2-2x-3y=0\,$

6.solution If the radius of the circle $\displaystyle x^2+y^2-8x+10y+k=0\,$ is 7,find the value of k.

7.solution If the equation $\displaystyle ax^2+2bxy-2y^2+8x+12y+6=0\,$ represents a circle,find the values of a and b.

8.solution Find the position of the point (3,1) with respect to $\displaystyle x^2+y^2-2x-4y+3=0\,$

9.solution Find the power of the point (2,-1) with respect to $\displaystyle 3x^2+3y^2+4x+2y+6=0\,$

10.solution Find the power of the point (a+b,a-b) with respect to $\displaystyle x^2+y^2+2bx-3b^2=0\,$

11.solution Find the equation to the point circle with centre (-2,3).

12.solution Find the equation to the circle passing through (0,0) and concentric with $\displaystyle x^2+y^2-2x-3y-4=0\,$
13.solution One end of the diameter of the circle$\displaystyle x^2+y^2-10x-2y+6=0\,$ is (3,5).Find the other end of the diameter.

14.solution Find the equation to the circle on the line segment joining the following points as diameter i).$\displaystyle (5,1),(7,5)\,$ ii).$\displaystyle (-4,-2),(1,1)\,$

15.solution Find the equation to the circle passing through the points$\displaystyle (1,3),(0,-2),(-3,1)\,$

16.solution Find the equation to the circle passing through the points$\displaystyle (a,0),(0,b),(a,b)\,$

17.solution Show that the points$\displaystyle (-6,0),(-2,2),(-2,-8),(1,1)\,$ are concyclic.

18.solution Find the circle which passes through (-1,2),(3,-2) and has its centre on the line $\displaystyle x-2y=0\,$

19.solution Find the circle which passes through (4,-3),(-1,2) and has its centre on the line $\displaystyle 3x+4y+1=0\,$

20.solution Find the length of the chord $\displaystyle x+2y=5\,$ of the circle $\displaystyle x^2+y^2=9\,$

21. i).The equation of the circumcircle of the triangle formed by the line $\displaystyle ax+by+c=0\,$ with the coordinate axes is $\displaystyle ab(x^2+y^2)+c(bx+ay)=0\,$

ii).If the two lines$\displaystyle a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0\,$ meet the coordinate axes in four distinct points ,then those points are concyclic if $\displaystyle a_1a_2=b_1b_2\,$

iii). If the two lines$\displaystyle a_1x+b_1y+c_1=0,a_2x+b_2y+c_2=0\,$ meet the coordinate axes in four distinct points ,then the equation to the circle passing through those points is $\displaystyle (a_1x+b_1y+c_1)(a_2x+b_2y+c_2)-(a_1b_2+a_2b_1)xy=0\,$

iv).If L=0 is a straight line intersecting the circle S=0,then the equation of the circle passing through the points of intersection is $\displaystyle S+\lambda L=0\,$ where L is a parameter.

21.solution Find the equation to the circumcircle of the traingle formed by the line 7x-3y-2=0 with the coordinate axes.

22.solution Show that the lines $\displaystyle 2x-5y+1=0,6y-15x+2=0\,$ intersect the coordinate axes in concyclic points. Also find the equation of the circle passing through those points.

23.solution Show that the pair of straight lines $\displaystyle ax^2+2hxy+ay^2+2gx+2fy+c=0\,$ meet the coordinate axes in concyclic points.Also find the equation of the circle through those cyclic points.

24.solution Find the equation of the circle which passes through the points of intersection of $\displaystyle x^2+y^2-x-y=0\,$ and $\displaystyle x+y=1\,$ and also through the point (1,1).

25.solution Find the equation of the circle which passes through the points of intersection of $\displaystyle x^2+y^2+3x-4y+5=0\,$ and $\displaystyle x-y+2=1\,$ and also through the point (2,3).

26.solution Find the equation to the circle passing through the points of intersection of the circle $\displaystyle x^2+y^2-8x-2y+7=0\,$ and the line $\displaystyle 4x+12y+1=0\,$ and which has its centre on y-axis.

27.solution Find the equation to the circle passing through the points (1,-2),(4,-3) and having the centre on the line $\displaystyle 3x+4y=7\,$

28.solution Find the equation of the circle which passes through (1,1),(2,2) and whose radius is unity.

29.solution Find the equation to the circle on AB as diameter and hence find the circle passing through $\displaystyle A(1,1),B(2,-1),C(3,2)\,$

30.solution Find the equation to the circle on AB as diameter and hence find the circle passing through $\displaystyle A(3,4),B(3,-6),C(-1,2)\,$

31.solution Find the equations of the tangents from the point(0,1) to the circle $\displaystyle x^2+y^2-2x-6y+6=0\,$

32.solution Find the locus of the point from which the lengths of the tangents to the circles $\displaystyle x^2+y^2+4x+3=0\,$ and $\displaystyle x^2+y^2-6x+5=0\,$ are in the ratio 2:3.

33.solution Find the equations of the tangents to the circle $\displaystyle x^2+y^2-4x-6y-12=0\,$ and parallel to $\displaystyle 4x-3y=1\,$ .

34.solution Find the equations of the tangents to the circle $\displaystyle x^2+y^2-2x-4y-4=0\,$ and parallel to $\displaystyle 3x-4y-1=0\,$ .

35.solution Find the equation of the circle which passes through (1,-2),(3,-4) and touches the X-axis.

36.solution Prove that the locus of a point tangents from which to the circle $\displaystyle x^2+y^2=a^2\,$ are inclined at an angle alpha is $\displaystyle (x^2+y^2-2a^2)^2 (\tan \alpha)^2=4a^2(x^2+y^2-a^2)\,$ .

37.solution Find the equations of circles which touch the axis of x at the origin and the line $\displaystyle 4x-3y+24=0\,$

38.solution Find the locus of point of intersection of two perpendicular tangents to the circle $\displaystyle x^2+y^2=a^2\,$

39.solution Show that the line x+y+1=0 touches the circle $\displaystyle x^2+y^2-3x+7y+14=0\,$ and find the point of contact.

40.solution Show that the line 3x=y+13 touches the circle $\displaystyle x^2+y^2-4x-6y+3=0\,$ and find the point of contact.

41.solution Prove that the tangent to the circle $\displaystyle x^2+y^2=5\,$ at (1,-2) also touches the circle $\displaystyle x^2+y^2-8x+6y+20=0\,$ and find the point of contact.

42.solution Find the equation of the tangent at(1,2) to the circle $\displaystyle x^2+y^2+2x-2y-3=0\,$ . Find also the equation of the tangent parallel to the above tangent.

43.solution Find the equation to the circle passing through the points of intersection of the lines x+2y-4=0 and the circle $\displaystyle x^2+y^2=4\,$ and touching the line x+2y=5.

44.solution Find the equation to the circle passing through the points of intersection of the lines x+2y-1=0 and the circle $\displaystyle x^2+y^2-2x+1=0\,$ and touching the line 2x-y+3=0.

45.solution Find the equation of the circle with centre on the line 2x+y=0 and which touches the lines 4x-3y+10=0 and 4x-3y-30=0.

46.solution Find the equation of the chord of contact of (4,-1) with respect to the circle $\displaystyle 2x^2+2y^2=11\,$ .

47.solution Find the pole of the line 3x+4y-45=0 with respect to the circle $\displaystyle x^2+y^2-6x-8y+5=0\,$

48.solution Show that the lines 2x+3y-12=0 and 3x+2y-2=0 are conjugate lines with respect to the circle $\displaystyle x^2+y^2=2\,$

49.solution What is the value of k if (4,k) and (2,3) are conjugate points with respect to the circle $\displaystyle x^2+y^2=17\,$

50.solution Find the value of k if the points (4,2) and (8,k) are conjugate with respect to $\displaystyle 3x^2+3y^2-12x+4y-4=0\,$

51.solution Find the value of k if the lines 2x+3y-4=0 and kx+4y-2=0 are conjugate with respect to $\displaystyle x^2+y^2=4\,$

52.solution Show that the poles of tangents of the circle $\displaystyle (x-p)^2+y^2=b^2\,$ with respect to the circle $\displaystyle x^2+y^2=a^2\,$ lie on the curve $\displaystyle (px-a^2)^2=b^2(x^2+y^2)\,$

53.solution Find the locus of the point whose polars with respect to the circles $\displaystyle x^2+y^2-4x-4y-8=0\,$ and $\displaystyle x^2+y^2-2x+6y-2=0\,$ are mutually perpendicular.

54.solution Show that the locus of the poles of the tangents to the circle $\displaystyle x^2+y^2=a^2\,$ with respect to the circle $\displaystyle (x+a)^2+y^2=2a^2\,$ is $\displaystyle y^2+4ax=0\,$

55.solution Write down the equation of the chord of the circle $\displaystyle x^2+y^2-2x-4y-20=0\,$ bisected at the point (2,0).

56.solution Find the equation of the chord of the circles i). $\displaystyle x^2+y^2=25\,$ having $\displaystyle (1,-1)\,$ as its midpoint. ii).$\displaystyle x^2+y^2=81\,$ having $\displaystyle (-2,-3)\,$ as its midpoint.

57.solution Find the equation of the chord of the circle $\displaystyle x^2+y^2+4x-2y-20=0\,$ having $\displaystyle (\frac{1}{2},\frac{-7}{2})\,$ as its midpoint.

58.solution Find the midpoint of the chord $\displaystyle 3x-y-1=0\,$ with respect to the circle $\displaystyle x^2+y^2-4x+6y+1=0\,$

59.solution Find the middle point of the chord of the circle $\displaystyle x^2+y^2=9\,$ intercepted by the line $\displaystyle 2x+y-5=0\,$

60.solution Find the mid point of the chord of the circle $\displaystyle x^2+y^2+4x+6y+2=0\,$ intercepted by the line $\displaystyle 3x+5y=13\,$

61.solutionFind the equation of the chord of the circle $\displaystyle x^2+y^2=15\,$ having mid point (3,-2). Also find the pole of that chord with respect to the circle.

62.solution Find the equation of the chord of the circle $\displaystyle x^2+y^2=25\,$ having mid point (1,2). Also find the pole of that chord with respect to the circle.

63.solution Find the locus of the midpoints of chords of the circle $\displaystyle x^2+y^2=r^2\,$ ,subtending a right angle at the point (a,b).

64.solution Find the equation of the tangents drawn from the origin to the circle $\displaystyle x^2+y^2+10x+10y+40=0\,$

65.solution Find the equation to the pair of tangents drawn from $\displaystyle (10,4)\,$ to the circles$\displaystyle x^2+y^2=25\,$

66.solution Show that the pair of tangents drawn from $\displaystyle (g,f)\,$ to the circles$\displaystyle x^2+y^2+2gx+2fy+c=0\,$ are at right angles if $\displaystyle g^2+g^2+c=0\,$

67.solution Find the angle between the pair of tangents drawn from (1,3) to the circles $\displaystyle x^2+y^2-2x+4y-11=0\,$

68.solution Tangents are drawn to the circle $\displaystyle x^2+y^2=a^2\,$ from a point which always lies on the line $\displaystyle lx+my=1\,$ . Prove that the locus of the mid-point of the chords of contact is $\displaystyle x^2+y^2-a^2(lx+my)=0\,$ .

69.solution Find the equation of the pair of tangents drawn from the point $\displaystyle (4,3)\,$ to the circle $\displaystyle x^2+y^2-2x-4y=0\,$ and hence find the angle between them.

70.solution Find the condition that the pair of tangents from the origin to the circle $\displaystyle x^2+y^2+2gx+2fy+c=0\,$ may be at right angles.

71.solution State whether the following pair of circles intersect or do not intersect or touch each other. $\displaystyle (x-1)^2+(y-3)^2=1\,$ and $\displaystyle (x+3)^2+(y-1)^2=9\,$

72.solution If the polar of the point $\displaystyle (x_1,y_1)\,$ w.r.t the circle $\displaystyle x^2+y^2=a^2\,$ touches the circle $\displaystyle (x-a)^2+y^2=a^2\,$ , show that the point lies on the curve $\displaystyle y^2+2ax=a^2\,$

73.solution The polar of P w.r.t the circle $\displaystyle x^2+y^2=a^2\,$ touches the circle $\displaystyle (x-f)^2+(y-g)^2=b^2\,$ .Prove that its locus is given by the equation $\displaystyle (fx+gy-a)^2=b^2(x^2+y^2)\,$ .

74.solution Find the condition that the two circles $\displaystyle x^2+y^2+2a_1 x+2b_1 y=0\,$ and $\displaystyle x^2+y^2+2a_2 x+2b_2 y=0\,$ touch each other.

75.solution Find the equation of the common chord of the circles $\displaystyle x^2+y^2+4x-12y+14=0\,$ and$\displaystyle x^2+y^2-14x+6y-22=0\,$ . Find the points of intersection of the circles.Also find the length of the common chord.

76.solution Find the locus of the poles of the line$\displaystyle \frac{x}{a}+\frac{y}{b}=1\,$ w.r.t the circles which touch the coordinate axes and whose centre lies in the first quadrant.

77.solution Show that the circle $\displaystyle x^2+y^2-4x-6y-12=0,x^2+y^2+6x+18y+26=0\,$ touch each other and find the point of contact.

78.solution If the two circles $\displaystyle x^2+y^2+2gx+2fy=0,x^2+y^2+2g_1 x+2f_1 y=0\,$ touch each other,prove that $\displaystyle fg_1=f_1 g\,$

79.solution Find the equations of the direct common tangents to the circles $\displaystyle x^2+y^2=1,(x-1)^2+(y-3)^2=4\,$

80.solution Find the equation of the pair of direct common tangents to the following circles. $\displaystyle x^2+y^2=16,x^2+y^2-4x-2y-4=0\,$

81.solution Find the equations to the transverse common tangents of the circles $\displaystyle x^2+y^2+4x+2y-4=0,x^2+y^2-4x-2y+4=0\,$ .

82.solution Find the equations to the transverse common tangents of the circles $\displaystyle x^2+y^2=1,(x-1)^2+(y-3)^2=4\,$ .

83.solution Find the equations of common tangents to the circles $\displaystyle x^2+y^2-4x-10y+28=0,x^2+y^2+4x-6y+4=0\,$ .

84.solution Write down the equation of the common tangent if the two circles $\displaystyle x^2+y^2+6x-6y+2=0,x^2+y^2-2x=0\,$ touch each other.

85.solution Show that the circles $\displaystyle x^2+y^2+2ax+c=0,x^2+y^2+2by+c=0\,$ touch each other if $\displaystyle \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}\,$ .

86.solution Find the length of the common chord of the two circles $\displaystyle (x-a)^2+y^2=a^2,x^2+(y-b)^2=b^2\,$ .

87.solution Find the length of the common chord of the two circles $\displaystyle x^2+y^2+4x-12y+14=0,x^2+y^2-14x+6y-22=0\,$

88.solution Prove that the length of the common chord of the circles $\displaystyle (x-a)^2+(y-b)^2=c^2,(x-b)^2+(y-a)^2=c^2\,$ is $\displaystyle \sqrt{[4c^2-2(a-b)^2]}\,$ . Hence find the condition that the circles may touch.

89.solution Find the equation to the circle whose diameter is the common chord of two circles $\displaystyle x^2+y^2-4x+6y-12=0,x^2+y^2+2x-2y-23=0\,$ . Hence find the length of the common chord.

90.solution Find the equation of the circle described on the common chord of the circles $\displaystyle x^2+y^2-4x-6y-12=0,x^2+y^2+6x+4y-12=0\,$ as diameter.

91.solution Find the equation of the circle having the common chord of the circles $\displaystyle x^2+y^2+x+5y-2=0,x^2+y^2+2x+5y-4=0\,$ as diameter.

92.solution Show that the length of the common of the two circles $\displaystyle x^2+y^2+2gx+c=0\,$ and $\displaystyle x^2+y^2+2fy-c=0\,$ is $\displaystyle 2\sqrt{\frac{(g^2-c)(f^2+c)}{g^2+f^2}}\,$

93.solution Find the equation of the circle which passes through the points of intersection of $\displaystyle x^2+y^2-2x+1=0,x^2+y^2-5x-6y+4=0\,$ and touch the line $\displaystyle 2x-y+3=0\,$

94.solution Find the equation of the circle which passes through the points of intersection of $\displaystyle x^2+y^2=4,x^2+y^2-2x-4y+4=0\,$ and touch the line $\displaystyle x+2y=5\,$

95.solution Find the equation of the circle whose radius is 5units and which touches the circle $\displaystyle x^2+y^2-2x-4y-20=0\,$ at the point (5,5).

96.solution Find the equation of the circle of radius $\displaystyle 2\sqrt{2}\,$ and passing through the points of intersection of the circles $\displaystyle x^2+y^2-2x-4y+4=0,x^2+y^2=4\,$ .

97.solution Find the equation of the circle which touches the line $\displaystyle x=y\,$ at the origin and passes through the point $\displaystyle (2,1)\,$ .

98.solution Find the angle between the circles $\displaystyle x^2+y^2+2x+4y+1=0\,$ and $\displaystyle x^2+y^2+4x+2y+4=0\,$

99.solution Find the acute angle of intersection of the following circles. $\displaystyle (x-2)^2+y^2=2,(x-2)^2+(y-1)^2=1\,$

100.solution If the circles $\displaystyle x^2+y^2-8x-6y+21=0\,$ and $\displaystyle x^2+y^2-2y+c=0\,$ cut each other orthogonally, find the value of c.

101.solution If the circles $\displaystyle x^2+y^2-4x-4y=0\,$ and $\displaystyle x^2+y^2-kx+2y+4=0\,$ cut each other orthogonally, find the value of k.

102.solution Find the equation passing through the origin and cutting the circles $\displaystyle x^2+y^2-8y+12=0,x^2+y^2-4x-6y-9=0\,$ orthogonally.

103.solution Find the equation passing through the origin and cutting the circles $\displaystyle x^2+y^2-4x-6y-3=0,x^2+y^2-16x-6y+4=0\,$ orthogonally.

104.solution Find the equation passing through the origin and which has its centre on the line$\displaystyle x+y=4\,$ and cuts circle $\displaystyle x^2+y^2-4x+2y+4=0\,$ orthogonally.

105.solution Find the equation of the circle which cut orthogonally the circles $\displaystyle x^2+y^2-6y+1=0,x^2+y^2-4y+1=0\,$ and touch the line $\displaystyle 3x+4y+5=0\,$

106.solution Prove that the two circles which pass through the points $\displaystyle (0,a),(0,-a)\,$ and touch the line $\displaystyle y=mx+c\,$ will cut orthogonally if $\displaystyle c^2=a^2(2+m^2)\,$

107.solution Find the equation of the circle which cuts orthogonally three circles $\displaystyle x^2+y^2+2x+4y+1=0,2x^2+2y^2+6x+8y-3=0\,$ and $\displaystyle x^2+y^2-2x+6y-3=0\,$

108.solution Find the equation of the circle which is orthogonal to each of the circles $\displaystyle x^2+y^2-x+y=0,2x^2+2y^2+5x-3y-2=0,x^2+y^2+4x-2y-2=0\,$

109.solution Find the equation of the circle which is orthogonal to each of $\displaystyle x^2+y^2-4x-2y+6=0,x^2+y^2-2x+6y=0,x^2+y^2-12x+2y+30=0\,$

110.solution Find the circle which passes through the points of intersection of the circles $\displaystyle x^2+y^2+6x+4y-12=0,x^2+y^2-4x-6y-12=0\,$ and cuts the circle $\displaystyle x^2+y^2-2x+3=0\,$ orthogonally.

111.solution Find the equation of the circle which is orthogonal to $\displaystyle x^2+y^2+2x+8=0,x^2+y^2-8x+8=0\,$ and which touches the line $\displaystyle x-y+4=0\,$

112.solution Find the equation to the radical axis of the two circles $\displaystyle x^2+y^2+10x-4y-1=0,x^2+y^2+5x+y+4=0\,$

113.solution Find the equation of the radical axis of the circles $\displaystyle 3x^2+3y^2-7x+8y+11=0,x^2+y^2-3x-4y+5=0\,$

114.solution Find the radical centre of the circles $\displaystyle x^2+y^2-4x-2y+6=0,x^2+y^2-2x+6y=0,x^2+y^2-12x+2y+30=0\,$

115.solution Find the radical centre of the circles $\displaystyle x^2+y^2-2x=0,x^2+y^2+2x=0,x^2+y^2-2y=0\,$

116.solution Find the radical centre of the circles $\displaystyle x^2+y^2=9,x^2+y^2-2x-2y-5=0,x^2+y^2+4x+6y-19=0\,$

117.solution Find the equation to the circle which is orthogonal to each of $\displaystyle x^2+y^2+2x+17y+4=0,x^2+y^2+7x+6y+11=0,x^2+y^2-x+22y+3=0\,$

118.solution Find the equation to the circle which is orthogonal to each of $\displaystyle x^2+y^2+4x+2y+1=0,2x^2+2y^2+8x+6y-3=0,x^2+y^2+6x-2y-3=0\,$

119.solution Find the equation to the circle passing through (1,-1) and belonging to the coaxal system determined by the circles $\displaystyle x^2+y^2-6x-6y+4=0,x^2+y^2-x+2y-3=0\,$

120.solution Find the equation of the circle belonging to the coaxal system determined by the circles $\displaystyle x^2+y^2-4x-6y-12=0,x^2+y^2+6x+4y-12=0\,$ and cuts the circle $\displaystyle x^2+y^2-2x+3=0\,$ orthogonally.

121.solution Find the equation to the circle touching the line $\displaystyle 2x-y+3=0\,$ and belonging to the coaxal system determined by $\displaystyle x^2+y^2-2x+1=0\,$ and the radical axis $\displaystyle x+2y-1=0\,$

122.solution The line $\displaystyle x+2y-1=0\,$ is the radical axis and the circle $\displaystyle x^2+y^2=1\,$ is a member of a coaxal system.Find the circle touching the line $\displaystyle x+2y=0\,$ and belonging to the system.

123.solution Find the limiting points of the coaxal system determined by the circles $\displaystyle x^2+y^2+5x+y+4=0,x^2+y^2+10x-4y-1=0\,$

124.solution Find the limiting points of the coaxal system determined by the circles $\displaystyle x^2+y^2+2x+4y+7=0,x^2+y^2+4x+2y+5=0\,$

125.solution Find the limiting points of the coaxal system determined by the circles $\displaystyle x^2+y^2-6x-6y+4=0,x^2+y^2-2x-4y+3=0\,$

126.solution If (1,2) and(3,1) are the limiting points of a coaxal system of circles find the radical axis.

127.solution (2,1) is one limiting point of a coaxal system of which the radical axis is $\displaystyle x+y+4=0\,$ .Find the other limiting point.

128.solution Find the other limiting point of the coaxal system of which one limiting point is (3,1) and radical axis is$\displaystyle x-y+2=0\,$

129.solution Find the equation of the circle which belongs to the coaxal system determined by (0,-3) and (-2,-1) and which is orthogonal to the circle $\displaystyle x^2+y^2+2x+6y+1=0\,$

130.solution Find the equation of a circle which passes through the origin and belongs to the coaxal system of which (1,2) (4,3)are the limiting points.

131.solution Find the equation of the circle belonging to the coaxal system of which the limiting points are $\displaystyle (2,-3),(0,-4)\,$ and which passes through (2,-1)

132.solution Tangents are drawn parallel to the line $\displaystyle y=mx\,$ to touch the circles of the coaxal system $\displaystyle x^2+y^2+2\lambda x+c=0\,$ . Show that the locus of their points of contact is the curve $\displaystyle x^2+2mxy-y^2=c\,$

133.solution Find the equation to the system of circles orthogonal to the coaxal system $\displaystyle x^2+y^2+3x+4y-2+k(x+y-7)=0\,$

134.solution Find the coaxal system which is orthogonal to the coaxal sytem $\displaystyle x^2+y^2+2x-3y-1+k(x-4y+1)=0\,$

135.solution Show that as k varies the circles$\displaystyle x^2+y^2+2ax+2by+2k(ax-by)=0\,$ form coaxal system.Find the radical axis.

136.solution The origin is a limiting point of a system of coaxal circles of which $\displaystyle x^2+y^2+2gx+2fy+c=0\,$ is a member.Show that the equations of the circles of the orthogonal system are $\displaystyle (x^2+y^2)(g+kf)+c(x+ky)=0\,$ for different values of k.

## Plane

### The Parabola

1.solution Write the equation of the parabola whose focus is (1,2) and directrix is $\displaystyle x+1=0\,$

2.solution Write the equation of the parabola whose focus is (-1,1) and directrix is $\displaystyle 4x+3y+2=0\,$

3.solution Determine the equation of the parabola with vertex at (6,2), its axis parallel to Y-axis and passes through (2,4).

4.solution Find the focus of the parabola i).$\displaystyle (y-1)^2=8(x-3\,$ . ii).$\displaystyle (y-2)^2=8(x-3)\,$ iii).$\displaystyle y^2-x-2y+2=0\,$

5.solution Find the equation of the parabola whose axis is parallel to the Y-axis and which passes through the points $\displaystyle (4,5),(-2,11),(-4,21)\,$

6.solution Find the equation of the parabola whose axis is parallel to X-axis and which passes through $\displaystyle (3,3),(6,5),(6,-3)\,$

7.solution Find the equation of the parabola whose axis is parallel to X-axis and passing through the points $\displaystyle (-1,3),(-2,1),(1,2)\,$

8.solution Find the equation of the parabola whose axis is parallel to Y-axis and passing through $\displaystyle (2,5),(1,4),(-1,8)\,$

9.solution Find the equation of the parabola whose focus is (3,-4) and directrix is $\displaystyle x+y+7=0\,$

10.solution Obtain the equation of the parabola whose focus is (4,5) and vertex is (3,6)

11.solution Find the vertex,latusrectum,axis,tangent at the vertex,focus and directrix of the parabola $\displaystyle x^2+8x+12y+4=0\,$

12.solution Find the vertex,latusrectum,axis,tangent at the vertex,focus and directrix of the parabola $\displaystyle x^2-2x-4y-3=0\,$

13.solution Find the equation of the tangent to the parabola i)$\displaystyle y^2=16x\,$ inclined at 60 degrees to X-axis. ii).$\displaystyle y^2=8ax\,$ at $\displaystyle (2a,4a)\,$

14.solution Find the equation of the normal to the parabola i).$\displaystyle y^2=8x\,$ at $\displaystyle (2,4)\,$ ii).$\displaystyle y^2=4x\,$ at $\displaystyle (1,2)\,$ iii).$\displaystyle y^2=4x\,$ whose slope is 2

15.solution The line $\displaystyle 2x-y+2=0\,$ touches the parabola $\displaystyle y^2=4px\,$ .Find p and also the point of contact.

16.solution Find the value of p if the line $\displaystyle x+y+2=0\,$ touches the parabola $\displaystyle y^2=px\,$

17.solution Find the condition if $\displaystyle px+qy+r=0\,$ is a tangent to $\displaystyle y^2=3ax\,$

18.solution If the line $\displaystyle y=mx+c\,$ is a tangent to the parabola $\displaystyle y^2=4a(x+a)\,$ prove that the condition is $\displaystyle c=am+\frac{a}{m}\,$

19.solution Show that the line $\displaystyle x-y+2=0\,$ is a tangent to $\displaystyle y^2=8x\,$ .Find the point of contact.

20.solution Show that the equation of common tangents to the circle $\displaystyle x^2+y^2=2a^2\,$ and the parabola $\displaystyle y^2=8ax\,$ is $\displaystyle y=\pm (x+2a)\,$

21.solution Find the equations of common tangents to the circle $\displaystyle x^2+y^2=8\,$ and to the parabola $\displaystyle y^2=16x\,$

22.solution Show that the locus of the point of intersection of perpendicular tangents to the parabola $\displaystyle y^2=4ax\,$ is the directrix $\displaystyle x+a=0\,$

23.solution Show that the equation of the chord joining the points $\displaystyle (x_1,y_1),(x_2,y_2)\,$ on the parabola $\displaystyle y^2=4ax\,$ is $\displaystyle (y-y_1)(y-y_2)=y^2-4ax\,$

24.solution Show that the locus of the foot of the perpendicular from the focus to the tangent of the parabola $\displaystyle y^2=4ax\,$ is $\displaystyle x=0\,$ ,the tangent to the vertex.

25.solution Find the equation to the pair of tangents to the parabola $\displaystyle y^2=6x\,$ which pass through $\displaystyle (-1,2)\,$

26.solution If a chord of the parabola $\displaystyle y^2=4ax\,$ touches the parabola $\displaystyle y^2=4bx\,$ . Show that the tangents at its extremities meet on the parabola $\displaystyle by^2=4a^2x\,$

27.solution Find the locus of the midpoints of chords of the parabola $\displaystyle y^2=4ax\,$ which subtend a right angle at the vertex of the parabola.

28.solution Show that the locus of the midpoints of chords of $\displaystyle y^2=4ax\,$ which subtend a constant angle alpha at the vertex is $\displaystyle [y^2-2ax+8a^2]^2=16a^2\cot ^{2}\alpha(4ax-y^2)\,$ .

29.solution Prove that the locus of the midpoints of the focal chords of the parabola $\displaystyle y^2=4ax\,$ is another parabola whose vertex is the focus of $\displaystyle y^2=4ax\,$ .

30.solution Show that the locus of the poles of chords which are normal to the parabola $\displaystyle y^2=4ax\,$ is $\displaystyle y^2(x+2a)+4a^3=0\,$

31.solution Show that the locus of the poles of the chords of the parabola $\displaystyle y^2=4ax\,$ which subtend a constant angle alpha at the vertex is the curve $\displaystyle (x+4a)^2\tan^2\alpha=4(y^2-4ax)\,$

32.solution Show that the locus of the poles of chords of the parabola $\displaystyle y^2=4ax\,$ which subtend a right angle at the vertex is$\displaystyle x+4a=0\,$

33.solution Show that the locus of poles of chords of the parabola $\displaystyle y^2=4ax\,$ which are at a constant distance 'a' from the focus is $\displaystyle y^2=4x(2a+x)\,$

33.solution The chord of contact of tangents from a point P to the parabola $\displaystyle y^2=4ax\,$ touches the circle $\displaystyle x^2+y^2=b^2\,$ .Prove that the locus of P is$\displaystyle 4a^2x^2=b^2(y^2+4a^2)\,$

34.solution Show that the locus of the midpoints of chords of the prabola$\displaystyle y^2=4ax\,$ which touch the circle $\displaystyle x^2+y^2=a^2\,$ is $\displaystyle (y^2-2ax)^2=a^2(y^2+4a^2)\,$

35.solution Show that the locus of poles of chords of the parabola$\displaystyle y^2=4ax\,$ at a constant distance b from the vertex is $\displaystyle b^2 y^2+4a^2(b^2-x^2)=0\,$

36.solution The polar of P w.r.t the parabola $\displaystyle y^2=4ax\,$ touches the circle $\displaystyle x^2+y^2=4a^2\,$ . Find the locus of P.

37.solution Show that the locus of the poles of chords of the parabola $\displaystyle y^2=4ax\,$ which are at constant distance 'd' from the focus is $\displaystyle d^2(y^2+4a^2)=4a^2(x+a)^2\,$ .

38.solution Show that the locus of the midpoints of chords of the parabola $\displaystyle y^2=6x\,$ and which touch the circle $\displaystyle x^2+y^2+4x-12=0\,$ is $\displaystyle [y^2-3x-6]^2=16(y^2+9)\,$

39.solution A tangent to the parabola $\displaystyle y^2+4bx=0\,$ meets $\displaystyle y^2=4ax\,$ at P and Q. Prove that the locus of the midpoint of PQ is $\displaystyle y^2(2a+b)=4a^2x\,$

40.solution Prove that the locus of midpoints of chords of constant length 2l of the parabola $\displaystyle y^2=4ax\,$ is $\displaystyle (y^2-4ax)(y^2+4a^2)+4a^2 l^2=0\,$

41.solution If the normals at the points $\displaystyle t_1,t_2\,$ on the parabola $\displaystyle y^2=4ax\,$ meet on the parabola, prove that $\displaystyle t_1 t_2=2\,$

42.solution Prove that the locus of the point of intersection of two perpendicular normals to the parabola $\displaystyle y^2=4ax\,$ is the parabola $\displaystyle y^2=a(x-3a)\,$

43.solution A chord which is normal at "t" to the parabola $\displaystyle y^2=4ax\,$ subtends a right angle at the vertex. Then prove that $\displaystyle t=\pm \sqrt{2}\,$

44.solution Prove that the circle on a focal radius of a prabola,as diameter touches the tangent at the vertex.

45.solution The line $\displaystyle lx+my+na=0\,$ meets the parabola $\displaystyle y^2=4ax\,$ at P,Q. The lines joining P and Q to the focus meet the parabola in M,N.Show that the equation to MN is $\displaystyle nx-my+la=0\,$

46.solution Show that the locus of the point,two of the normals from which to the parabola $\displaystyle y^2=4ax\,$ are coincident is $\displaystyle 27ay^2=4(x-2a)^3\,$

47.solution From the points of $\displaystyle 2x-3y+4=0\,$ tangents are drawn to $\displaystyle y^2=4ax\,$ . Show that the chords of contact pass through a fixed point.

48.solution P is a point on the line $\displaystyle lx+my+n=0\,$ .The polar of P w.r.t the parabola $\displaystyle y^2=4ax\,$ meets the curve in Q and R.Show that the locus of the midpoint of QR is $\displaystyle l(y^2-4ax)+2a(lx+my+n)=0\,$

49.solution Show that the tangent at one extremity of a focal chord of a parabola is parallel to the normal at the other extremity.

50.solution Prove that the length of the chord of contact of tangents drawn from $\displaystyle (x_1,y_1)\,$ to the parabola $\displaystyle y^2=4ax\,$ is $\displaystyle \frac{\sqrt{(y_1^{2}-4ax_1)}\sqrt{(y_1^{2}+4a^2)}}{a}\,$ .

### The Ellipse

1.solution Find the eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse $\displaystyle 9x^2+16y^2=144\,$

2.solution Find the lengths of major axis, minor axis, latus rectum, eccentricity, centre, foci, equations of directrices of the ellipse $\displaystyle \frac{x^2}{64}+\frac{y^2}{36}=1\,$

3.solution Find the eccentricity,coordinates of focus,length of latus rectum and equations of directrices of the ellipse $\displaystyle 4x^2+y^2-8x+2y+1=0\,$

4.solution Find the lengths of major axis, minor axis, latus rectum, eccentricity, centre, foci, equations of directrices of the ellipse $\displaystyle x^2+2y^2-4x+12y+14=0\,$

5.solution Find the lengths of major axis, minor axis, latus rectum, eccentricity, centre, foci, equations of directrices of the ellipse $\displaystyle 3x^2+y^2-6x-2y-5=0\,$

6.solution If the two ends of major axis of an ellipse are (5,0),(-5,0). Find the equation of ellipse if its focus lies on the line $\displaystyle 3x-5y-9=0\,$

7.solution Find the equation of the ellipse in the usual form,if it passes through the points (-2,2) and (3,1).(axis are along the coordinate axes and centre at the origin).

8.solution Find the equation of the ellipse with a focus at(1,-1),e=2/3 and directrix is $\displaystyle x+y+2=0\,$

9.solution Find the eccentricity of the ellipse ,if its length of the latus rectum is equal to half of its major axis.

10.solution Find the equation of the ellipse referred to its major axis and minor axis as the axes of coordinates a and y axes respectively with latus rectum of length 4 and distance between foci $\displaystyle 4\sqrt{2}\,$

11.solution Find the equation of ellipse with length of latus rectum 15/2 and distance between foci 2.

12.solutionShow that the condition for a straight line $\displaystyle y=mx+c\,$ be a tangent to the ellipse$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ is $\displaystyle c^2=a^2 m^2+b^2\,$

13.solution If the length of the latus rectum is equal to half of its minor axis of an ellipse in the standard form,then find the eccentricity of the ellipse.

14.solution Find the equations of tangent and normal to the ellipse $\displaystyle 2x^2+3y^2=11\,$ at the point whose ordinate is 1.

15.solution Find the equation of the tangent and normal to the ellipse $\displaystyle x^2+2y^2-4x+12y+14=0\,$ at (2,-1).

16.solution Find the equation of tangents to the ellipse $\displaystyle 2x^2+y^2=8\,$ which is parallel to $\displaystyle x-2y-4=0\,$ .

17.solution Find the equations to the tangents to the ellipse $\displaystyle x^2+2y^2=3\,$ drawn from the point (1,2).

18.solution Show that the foot of the perpendicular drawn from the centre on any tangent to the ellipse lies on the curve $\displaystyle (x^2+y^2)^2=a^2 x^2+b^2 y^2\,$

19.solution If the normal at one end of a latus rectum of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ passes through one end of the minor axis,then show that $\displaystyle e^4+e^2=1\,$

20.solution Show that the points of intersection of the perpendicular tangents to an ellipse lies on a circle.

21.solution Find the equation of the tangents to the ellipse $\displaystyle 2x^2+y^2=8\,$ is perpendicular to $\displaystyle x+y+2=0\,$

22.solution Find the coordiantes of the points on the ellipse $\displaystyle x^2+3y^2=37 \,$ at which the normal is parallel to the line $\displaystyle 6x-5y=2\,$

23.solution Prove that the sum of the squares of the perpendiculars on any tangent of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ (a > b) from two points on the minor axis,each at a distance of $\displaystyle \sqrt{a^2-b^2}\,$ from the centre is $\displaystyle 2a^2\,$

24.solution Find the locus of the point of intersection of the two tangents to the ellipse $\displaystyle b^2 x^2+a^2 y^2=a^2 b^2\,$ ,which include an angle theta.

25.solution Find the pole of the line $\displaystyle x-y+2=0\,$ with w.r.t the ellipse $\displaystyle x^2+2y^2-4x+12y+14=0\,$

26.solution Find the pole of the line $\displaystyle 21x-16y-12=0\,$ w.r.t the ellipse $\displaystyle 3x^2+4y^2=12\,$

27.solution Find the pole of the line $\displaystyle 3x-5y-9=0\,$ w.r.t the ellipse $\displaystyle 4x^2+8y^2-16x+15=0\,$

28.solution Find the equation of a straight line through the point (2,1) and conjugate to the straight line $\displaystyle 9x+2y=1\,$ w.r.t the ellipse $\displaystyle 3x^2+2y^2=1\,$

29.solution Show that the two lines $\displaystyle 9x+2y=1,3x+2y=11\,$ are conjugate w.r.t the ellipse $\displaystyle 3x^2+2y^2=1\,$

30.solution Find the value of k,if the lines$\displaystyle x+y+k=0,3x-2y-7=0\,$ are conjugate w.r.t the ellipse $\displaystyle x^2+3y^2=9\,$

31.solution Find the value of k if $\displaystyle (1,2),(k,-1)\,$ are conjugate w.r.t the ellipse $\displaystyle 2x^2+3y^2=6\,$

32.solution Show that the poles of the tangents of $\displaystyle y^2=4kx(k > 0)\,$ w.r.t the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ lie on a parabola.

33.solution Show that the poles of normal chords of the ellipse$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ lie on the curve $\displaystyle \frac{a^6}{x^2}+\frac{b^6}{y^2}=(a^2-b^2)^2\,$

34.solution Show that the poles of the tangents to the circle $\displaystyle x^2+y^2=a^2+b^2\,$ w.r.t the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ lies on $\displaystyle \frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{a^2+b^2}\,$

35.solution Show that the poles of the tangents to the auxiliary circle w.r.t the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ is the curve $\displaystyle \frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{a^2}\,$

36.solution Prove that the product of the perpendicular from the foci on any tangent to the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ is equal to $\displaystyle b^2\,$

37.solution Find the equation of the pair of tangents to the ellipse $\displaystyle x^2+3y^2=3\,$ from the point $\displaystyle (2,-1)\,$ .

38.solution A chord PQ of an ellipse subtends a right angle at the centre of the ellipse $\displaystyle S=0\,$ .Show that the point of intersection of tangents at P and Q lies on the ellipse$\displaystyle \frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{a^2}+\frac{1}{b^2}\,$

39.solution Prove that the pair of tangents drawn to $\displaystyle 9x^2+16y^2=144\,$ are perpendicular to eachother.

40.solution Show that the equation of the auxiliary circle of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ is $\displaystyle x^2+y^2=a^2\,$

41.solution Tangents at right angles are drawn to the ellipse$\displaystyle S=0\,$ .Show that the locus of the midpoints of chords of contact is the curve $\displaystyle [\frac{x^2}{a^2}+\frac{y^2}{b^2}]^2=\frac{x^2+y^2}{a^2+b^2}\,$

42.solution Find the locus of the midpoints of chords of an ellipse,whose poles lie on the auxiliary circle.

43.solution P is a point on the ellipse $\displaystyle S=0\,$ and Q is its corresponding point on the auxiliary circle. Prove that the locus of the point of intersection of the normals at P and Q is the circle given by $\displaystyle x^2+y^2=(a+b)^2\,$

44.solution Find the equation of the ellipse whose vertices are $\displaystyle (-4,1),(6,1)\,$ and whose focus lies on the line $\displaystyle x-2y=2\,$

45.solution Find the equation of the ellipse whose vertices are $\displaystyle (-4,3),(8,3)\,$ and whose eccentricity is 5/6.

46.solution Find the point of contact of the tangent line $\displaystyle 4x+y-7=0\,$ to the ellipse $\displaystyle x^2+3y^2=3\,$

47.solution Find the value of k if the line $\displaystyle x+ky-5=0\,$ is a tangent to the ellipse $\displaystyle 4x^2+9y^2=20\,$

48.solution Find the value of k and hence the point of contact of the tangent line $\displaystyle 4x+y+k=0\,$ with the ellipse $\displaystyle x^2+3y^2=3\,$

49.solution Find the equations to the tangents to the ellipse $\displaystyle 4x^2+3y^2=5\,$ which are parallel to $\displaystyle 3x-y+7=0\,$

50.solution Show that the locus of poles of chords of ellipse $\displaystyle S=0\,$ which touch the parabola $\displaystyle y^2=4px\,$ is $\displaystyle pa^2 y^2+b^4 x=0\,$

### Hyperbola

1.solution Write down the equation to the hyperbola whose focus is $\displaystyle (4,0)\,$ ,directrix is the line $\displaystyle 4x-9=0\,$ and eccentricity is 4/3.

2.solution Find the equation to the hyperbola whose focus is $\displaystyle (1,2)\,$ eccentricity $\displaystyle \sqrt{3}\,$ and directrix is $\displaystyle 2x+y-1=0\,$

3.solution What is the equation to the hyperbola if the latusrectum is 9/2 and eccentricity is 5/4.

4.solution Obtain the equation of the hyperbola in standard form whose latusrectum is 4 and eccentricity is 3.

5.solution Determine the equation to the hyperbola whose centre is (0,0),distance between the foci is 18 and that between the directrices is 8.

6.solution A hyperbola has one focus at the origin and its eccentricity is $\displaystyle \sqrt{2}\,$ .One of its directrices is$\displaystyle x+y+1=0\,$ .Find the equation of the hyperbola.

7.solution Find the centre,eccentricity,length of latusrectum,foci,vertices and equations to the directrices of the hyperbola $\displaystyle \frac{(y-4)^2}{9}-\frac{(x-5)^2}{16}=1\,$

8.solution Find the centre,eccentricity,length of latusrectum,foci,vertices and equations to the directrices of the hyperbola $\displaystyle 9x^2-16y^2-18x-32y-151=0\,$

9.solution What are the coordinates of the foci of the hyperbola $\displaystyle \frac{(x+1)^2}{25}-\frac{(y+2)^2}{16}=1\,$

10.solution Write down the equations of the directrices of the hyperbola $\displaystyle \frac{x^2}{144}-\frac{y^2}{36}=1\,$

11.solution Show that the ellipse $\displaystyle 7x^2+16y^2=112\,$ and the hyperbola $\displaystyle \frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}\,$ have the same foci.

12.solution Find the equations to the tangents to the hyperbola $\displaystyle 3x^2-y^2=3\,$ which are perpendicular to $\displaystyle x+3y=2\,$

13.solution Find the equations of the tangents to the hyperbola $\displaystyle 3x^2-4y^2=12\,$ which make equal intercepts on the axes.

14.solution Find the value of k if the line $\displaystyle 3x-y=k\,$ is a tangent to $\displaystyle 3x^2-y^2=3\,$

15.solution Find the equations of tangents to the hyperbola $\displaystyle 4x^2-3y^2=24\,$ which make an angle of 60 degrees with X-axis.

16.solution Prove that the line $\displaystyle 21x+5y-116=0\,$ touches the hyperbola $\displaystyle 7x^2-5y^2=232\,$ and find the point of contact.

17.solution Show that the line $\displaystyle x\cos\alpha+y\sin\alpha=p\,$ touches the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ if $\displaystyle a^2 \cos^2 \alpha-b^2 \sin^2 \alpha=p^2\,$

18.solution Find the equations of the tangents to the hyperbola $\displaystyle 9x^2-16y^2=1\,$ drawn parallel to to the line $\displaystyle 9x+8y=10\,$

19.solution Find the equation of the normal at (1,0)on the hyperbola $\displaystyle x^2-4y^2=1\,$

20.solution Show that the locus of poles w.r.t hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ of tangents to the parabola $\displaystyle y^2=4ax\,$ is $\displaystyle a^3 y^2+b^4 x=0\,$

21.solution Show that the locus of the pole of any tangent to the circle $\displaystyle x^2+y^2=a^2\,$ w.r.t the hyperbola $\displaystyle x^2-y^2=a^2\,$ is the circle itself.

22.solution The polar of any point on the ellipse $\displaystyle S=0\,$ w.r.t the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ will touch the ellipse.

23.solution If the polar of a point w.r.t ellipse $\displaystyle S=0\,$ touch the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ ,then show that the locus of point is is the hyperbola.

24.solution Prove that the locus of points the polars of which w.r.t the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ touch the circle $\displaystyle x^2+y^2=c^2\,$ i f $\displaystyle \frac{x^2}{a^4}+\frac{y^2}{b^4}=\frac{1}{c^2}\,$

25.solution Show that the locus of the poles w.r.t the parabola $\displaystyle y^2=4ax\,$ of tangents to the hyperbola $\displaystyle 4x^2-3y^2=a^2\,$ is $\displaystyle 12x^2+y^2=3a^2\,$

26.solution Find the line passing through the point (-2,1) and conjugate to the line $\displaystyle 8x+3y-4=0\,$ w.r.t $\displaystyle 2x^2-y^2=1\,$

27.solution Find the equation to the line passing through (1,2) and conjugate to the line $\displaystyle 3x+2y+6=0\,$ w.r.t hyperbola $\displaystyle 3x^2-4y^2=12\,$

28.solution Show that the locus of poles w.r.t parabola $\displaystyle y^2=4ax\,$ of the tangents to the hyperbola $\displaystyle x^2-y^2=a^2\,$ is the ellipse $\displaystyle 4x^2+y^2=4a^2\,$

29.solution Show that the locus of the foot of perpendicular from the centre of the hyperbola $\displaystyle \frac{x_2}{a^2}-\frac{y^2}{b^2}=1\,$ on a variable tangent is $\displaystyle (x^2+y^2)^2=a^2 x^2-b^2 y^2\,$

30.solution Tangents to the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ make angles $\displaystyle \alpha,\beta\,$ with the transverse axis. Find the locus of their point of intersection if $\displaystyle \tan\alpha+\tan\beta=k\,$

31.solution Tangents drawn from $\displaystyle (\alpha,\beta)\,$ to the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ make an angle $\displaystyle \theta_1,\theta_2\,$ with the x-axis.If $\displaystyle \tan\theta_1 \tan\theta_2=1\,$ prove that$\displaystyle \alpha^2-\beta^2=a^2+b^2\,$

32.solution Find the equation of the chord of the hyperbola $\displaystyle 4x^2-y^2=4\,$ which is bisected at the point $\displaystyle (2,-3)\,$

33.solution Find the equation to the hyperbola whose asymptotes are $\displaystyle 3x=\pm5y\,$ and vertices are $\displaystyle (\pm5,0)\,$

34.solution The asymptotes of the hyperbola are parallel to the lines $\displaystyle 2x+3y =0,3x+2y=0\,$ . Its centre is at $\displaystyle (1,2)\,$ and passes thro' the point $\displaystyle (5,3)\,$ .Find its equation.

35.solution Find the equation of the hyperbola whose asymptotes are $\displaystyle 7x+5y-3=0,2x+4y+1=0\,$ and passing thro' $\displaystyle (-2,1)\,$

36.solution Find the equation of the hyperbola whose asymptotes are $\displaystyle 2x+3y-5=0,5x+3y-8=0\,$ and passing thro' $\displaystyle (1,-1)\,$

37.solution Find the locus of midpoints of the chords of the parabola $\displaystyle \frac{x^2}{9}-\frac{y^2}{4}=1\,$ which are parallel to $\displaystyle 3x+8y-4=0\,$

38.solution Show that the locus of midpoints of the chord of the hyperbola$\displaystyle x^2-y^2=a^2\,$ ,which touch the parabola $\displaystyle y^2=4ax\,$ is $\displaystyle y^2(x-a)=x^3\,$

39.solution Show that the locus of midpoints of the chord of the hyperbola$\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ ,which pass through the focus $\displaystyle (ae,0)\,$ is $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{ex}{a}\,$

40.solution P is any point on the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ whose vertex is A(a,0).Show that the locus of the middle point of AP is $\displaystyle \frac{(2x-a)^2}{a^2}-\frac{4y^2}{b^2}=1\,$

41.solution A tangent of the auxiliary circle of the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ intersects it in P and Q.Find the locus of midpoint of PQ.

42.solution Find the locus of midpoints of the chords of hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ drawn parallel to the line $\displaystyle y=mx\,$

43.solution Find the asymptotes of the hyperbola $\displaystyle 2x^2+5xy+2y^2-11x-7y-4=0\,$

44.solution Find the asymptotes of the hyperbola $\displaystyle 2x^2-xy-y^2+2x-2y+2=0\,$

45.solution If e1,e2 are the eccentricities of a hyperbola and its conjugate,prove that $\displaystyle \frac{1}{e_1^{2}}+\frac{1}{e_2^{2}}=1\,$

46.solution If a tangent at a point P to a hyperbola meets the asymptotes in Q and R show that P is the midpoint of QR.

47.solution Show that the points of intersection of the asymptotes of hyperbola with its directrices lie on the auxiliary circle.

48.solution Show that the midpoints of normal chords of $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ is $\displaystyle [\frac{a^6}{x^2}-\frac{b^6}{y^2}][\frac{x^2}{a^2}-\frac{y^2}{b^2}]^2=(a^2+b^2)^2\,$

49.solution Show that the locus of the foot of the perpendicular drawn from the centre of the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ on any normal to it is $\displaystyle (a^2 y^2-b^2 x^2)(x^2+y^2)^2=(a^2+b^2)x^2 y^2\,$

50.solution Prove that the product of the perpendiculars from any point on the hyperbola$\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\,$ to its asymptotes is constant.

51.solutionShow that the portion of any tangent to the hyperbola,intercepted between the asymptotes is bisected at the point of contact.

### Polar Coordinates

1.solution What are the polar coordinates of $\displaystyle (-1,-1)\,$

2.solution What are the cartesian coordinates of the points

 i). $\displaystyle (3,\frac{\pi}{6})\,$

ii). $\displaystyle (4,\frac{\pi}{3})\,$



3.solution Determine the lengths of the sides of the triangle whose vertices are $\displaystyle (-3,\frac{\pi}{6}),(5,\frac{5\pi}{6}),(7,\frac{7\pi}{6})\,$

4.solution Find the distance between the points below

i). $\displaystyle (-1,\frac{\pi}{8}),(\sqrt{2},\frac{3\pi}{8})\,$

ii). $\displaystyle (-3,45^\circ),(7,105^\circ)\,$

5.solution Find the area of the triangle formed by the points $\displaystyle (-3,-30^\circ),(5,150^\circ),(7,120^\circ)\,$

6.solution Find the area of the triangle formed by the points $\displaystyle (1,\frac{\pi}{6}),(2,\frac{\pi}{3}),(3,\frac{\pi}{2})\,$

7.solution Prove that the points $\displaystyle (\frac{3}{2},0),(-\sqrt{2},45^\circ),(-\frac{3}{5},90^\circ)\,$ are collinear.

8.solution Find the equation of the line joining the points $\displaystyle (3,\frac{3\pi}{4}),(2,\frac{\pi}{4})\,$

9.solution Find the equation of the line joining the points $\displaystyle (2,\frac{\pi}{6}),(3,\frac{\pi}{3})\,$

10.solution Find the equation of the line passing thro'the point $\displaystyle (2,\frac{\pi}{6})\,$ parallel and perpendicular to the line $\displaystyle \frac{3}{r}=4\cos\theta-3\sin\theta\,$

11.solution Find the equation of the line passing thro' $\displaystyle (2,\frac{\pi}{3})\,$ and parallel to $\displaystyle r(4\cos\theta+\sqrt{3}\sin\theta)=2\,$

12.solution Find the equation of the line passing thro' $\displaystyle (-1,\frac{\pi}{2})\,$ and parallel to $\displaystyle 4=r(2\cos\theta+\sqrt{3}\sin\theta)\,$

13.solution Find the length of the perpendicular from the origin on the line $\displaystyle \frac{8}{r}=\sqrt{3}\cos\theta+\sin\theta\,$ .Also determine the angle made by the perpendicular with the intial line.

14.solution Find the perpendicular distance from the origin to $\displaystyle 6=r(\cos\theta+\sqrt{3}\sin\theta)\,$