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

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1 Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x 6y + 4 = 0 2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3). (ii) State the values of a and b if x 2 + y 2 + ax + by + 9 = 0 represents a circle with centre (1, 3). 3. Find the equation of the circle with centre C and radius r given (a) C = ( 1, 8), r = 5 (b) C = ( 7, 3), r = 10 (c) C = (a, b), r = (a + b) 4. Find the centre and the radius of the circle (a) x 2 + y 2 + 6x + 8y 96 = 0 (b) x 2 + y 2 6x + 4y 12 = 0 (c) 5x 2 + 5y 2 2x 3y = 0 5. If the radius of the circle x 2 + y 2 8x + 10y + k = 0 is 7, find the value of k. 6. Find the equation to the point circle with centre (a) ( 2, 3) (b) (4, 3) (c) (0, 0) 7. (i) Write the equation of circle having (3, 4) and ( 7, 2) as the extremities of the diameter. (ii) Find the equation to the circle passing through the point (1, 2) and concentric with

2 x 2 + y 2 + 8x + 12y + 15 = (i) Find the equation to the circle passing through the points ( 2, 8), (7, 1) and (1, 2). (ii) Find the centre and radius of the circle passing through (5, 4), (4, 3), (7, 0). 9. Find the equation of the circle having the centre (a) ( 2, 3) and passing through the origin (b) (0, 0) and passing through (2, 1) (c) (2, 3) and passing through (0, 0) (d) (0, 0) and passing through (3, 4) 10. Find the position of the point (a) (3, 1) with respect to x 2 + y 2 2x 4y + 3 = 0 (b) (2, 3) w.r to x 2 + y 2 + 2x + 2y 7 = 0 (c) (1, 2) w.r. to x 2 + y 2 4x + 6y 3 = Find the equation to the circle passing through (a) (0, 0) and concentric with x 2 + y 2 2x 3y 4 = 0 (b) (1, 0) and concentric with x 2 + y 2 3y + 10 = 0 (c) (3, 4) and concentric with x 2 + y 2 + 4x 2y + 1 = An end of the diameter of the circle (a) x 2 + y 2 10x 2y + 6 = 0 is (3, 5). Find the other end of the diameter. (b) x 2 + y 2 2x + 4y = 0 is (3, 1). Find the other end of the diameter.

3 13. (i) Find the equation to the circle on the line segment joining the following points as diameter (a) (5, 1), (7, 5) (b) ( 4, 2), (1, 1) (c) ( 2, 1), (4, 3) (ii) Find the equation to the circle passing through the points (a) (1, 3), (0, 2), ( 3, 1) (b) (1, 1), (3, 2), (0, 4) (c) (a, 0), (0, b), (a, b) (iii) Show that the points ( 6, 0), ( 2, 2), ( 2, 8) and (1, 1) are concyclic. 14. Find the circle which passes through (a) ( 1, 2), (3, 2) and has its centre on the lines x 2y = 0 (b) (4, 3) and ( 1, 2) and having centre on the line 3x + 4y + 1 = The abscissae of two points A and B are the roots of the equation x 2 + 2ax b 2 = 0 and their ordinates are the roots of the equation y 2 + 2py q 2 = 0. Find the equation of the circle on AB as diameter. 16. Find the length of intercept made by the circle (a) x 2 + y 2 + 8x 12y 9 = 0 on y axis (b) x 2 + y 2 + 2gx + 2fy + c = 0 on x axis (c) 2x 2 + 2y 2 7x 2y + 3 = 0 on x axis 17. (i) What are the parametric equations of the circle (x + 5) 2 + (y + 6) 2 = 100? (ii) What is the equation of the circle on which (2 + 4 cos, 1 + 4sin ) is a point, being the parameter? (iii) If x = cos, y = 3 + 5sin, show that the locus of the point (x, y) is a circle. Find its centre and radius. 18. For any value of, prove that the locus of the point of intersection of the lines x cos + y sin = a and x sin y cos = b is a circle. Find the centre and the radius.

4 19. Find the equation of the tangent to the circle x 2 + y 2 2x 4y + 3 = 0 at (2, 3). 20. What is the length of the tangent from (5, 1) to the circle x 2 + y 2 + 6x 4y 3 = 0? 21. Find the equation to the tangent to the circle (a) x 2 + y 2 + 6x + 4y 39 = 0 at (3, 2) (b) x 2 + y 2 6x 3y 2 = 0 at (2, 2) (c) x 2 + y 2 + 2x 2y 3 = 0 at (1, 2) 22. Find k if the line (a) kx 3y + 10 = 0 is a tangent to x 2 + y 2 = 10 (b) 3x + 4y + k = 0 is a tangent to x 2 + y 2 = Find the equation of the circle with centre at (a) ( 2, 3) and touching x axis (b) (3, 2) and touching y axis (c) (12, 5) and touching the line 3x 4y + 9 = 0 (d) (4, 3) and touching the line 5x 12y 10 = If a line through P(3, 4) cuts the circle x 2 + y 2 = 13 in A and B, find PA. PB. 25. (i) Find the equation of the tangents to the circle (a) x 2 + y 2 4x + 6y 12 = 0 and parallel to x + y 8 = 0 (b) x 2 + y 2 4x 6y 12 = 0 and parallel to 4x 3y = 1 (ii) Show that the line

5 (a) x + y + 1 = 0 touches the circle x 2 + y 2 3x + 7y + 14 = 0 and find the point of contact. (b) 3x = y + 13 touches the circle x 2 + y 2 4x 6y + 3 = 0 and find the point of contact. 26. Find the area formed by the tangent at (x 1, y 1 ) to the circle x 2 + y 2 = a 2 with the coordinate axes. 27. Find the equations of the circle touching the (a) axis of x at the origin and the line 3y = 4x + 24 (b) the line x = y at the origin and passing through (2, 1) (c) line 2x 3y + 1 = 0 at (1, 1) and having the radius 13. (d) the axes in the first quadrant and the line 3x + 4y = 12. (e) line 3x 4y + 5 = 0 at (1, 2) and having the radius (i) Prove that the tangent to the circle x 2 + y 2 = 5 at (1, 2) also touches the circle x 2 + y 2 8x + 6y + 20 = 0 and find the point of contact. (ii) Find the equations of the circles which pass through the point ( 4, 3) and touch the lines x + y = 2 and x y = 2. (iii) The circle x 2 + y 2 4x + 4y 1 = 0 cuts the positive coordinates axes in A and B respectively. Find the equation to the diameter of the circle perpendicular to the chord AB. 29. (i) Find the power of the point (a) (2, 1) w.r. to 3x 2 + 3y 2 + 4x + 2y + 6 = 0 (b) ( 3, 5) w.r. to x 2 + y 2 6x + 4y 3 = 0 (c) (a + b, a b) with respect to x 2 + y 2 + 4x 2y + 1 = 0 (ii) Find the length of the chord (a) x + 2y = 5 of the circle x 2 + y 2 = 9 (b) 4x 3y 5 = 0 of the circle x 2 + y 2 + 3x y 10 = 0

6 30. (i) Find the intercept made by the circle x 2 + y 2 4x + 6y 5 = 0 on x axis. (ii) If the length of tangent from (2, 5) to the circle x 2 + y 2 5x + 4y + xk = 0 is 37 find k. (iii) Find the locus of the point from which the lengths of the tangents to the circles x 2 +y 2 + 4x + 3 = 0 and x 2 + y 2 6x + 5 = 0 are in the ratio 2 : 3. (iv) Find the length of the tangent from (a) (0, 5) to the circle x 2 + y 2 + 2x 4 = 0 (b) (5, 4) to the circle x 2 + y y = 0 (c) (3, 0) to the circle x 2 + y 2 + 4x 6y 9 = 0 (v) Find the value of k if the length of the tangent drawn from the point? (a) (5, 4) to x 2 + y 2 + 2ky = 0 is 1. (b) ( 1, 1) to x 2 + y 2 4x + k = 0 is (i) Find the equation to the normal from (2, 4) to the circle x 2 + y 2 = 16. (ii) Find the equation of the normal at (2, 3) to the circle x 2 + y 2 + 4x + 6y 39 = 0. Find the second point where the normal meets the circle. (iii) If a line through (3, 4) cuts the circle x 2 + y 2 = 13 in (2, 3), find the other point which this line cuts the circle. 32. Find the equations of the circles which touch the x axis at a distance 4 from the origin and makes an intercept 6 on the axis of y. 33. (i) Find the angle between the pair of tangents drawn from (1, 3) to the circle x 2 +y 2 2x + 4y 11 = 0. a 2. (ii) Find the locus of the point of intersection of two perpendicular tangents to the circle x 2 + y 2 = (iii) Find the equations of tangents drawn from (4, 2) to the circle x 2 + y 2 = 10. Show that the two tangents are at right angle.

7 (iv) Find the equation of the chord of contact of (4, 1) with respect to the circle 2x 2 + 2y 2 = 11. (v) Find the polar of ( 7, 9) with respect to the circle x 2 + y 2 12x 8y 48 = (i) What is the value of k if (4, k) and (2, 3) are conjugate points with respect to the circle x 2 + y 2 = 17. (ii) Find the value of k if 2x + 3y = 12 and kx + 2y = 2 are conjugate lines with respect to the circle x 2 + y 2 = 2. (iii) Find the pole of the line 3x + 4y 45 = 0 with respect to the circle x 2 + y 2 6x 8y + 5 = Show that the lines 2x + 3y 12 = 0 and 3x + 2y 2 = 0 are conjugate lines with respect to the circle x 2 + y 2 = (i) Find the inverse point of (h, k) with respect to the circle x 2 + y 2 = a 2. (ii) Find the inverse point of (3, 2) with respect to the circle x 2 + y 2 4x + 6y 3 = Find the inverse point of the origin with respect to the circle x 2 + y 2 + 2gx + c = (i) Find the locus of the point whose polars with respect to the circles x 2 + y 2 4x 4y 8 = 0 and x 2 + y 2 2x + 6y 2 = 0 are mutually perpendicular. (ii) If (x 1, y 1 ) is the pole of the line lx + my + n = 0 with respect x 2 + y 2 + 2gx + 2fy + c = 0 then 2 x1 g y1 f r where r 2 = g 2 + f 2 c. m g mf n 39. Find the equation of the chord of contact of the point: (a) (2, 3) w.r. to the circle x 2 + y 2 = 5 (b) (1, 1) w.r. to the circle x 2 + y 2 = 1

8 40. (i) Find the equation to the polar of the point: (a) (3, 4) w.r. to the circle x 2 + y 2 = 25 (b) ( 2, 3) w.r. to the circle x 2 + y 2 4x 6y + 5 = 0 (c) (1, 2) w.r. to the circle x 2 + y 2 = 7 (d) (3, 1) w.r. to the circle 2x 2 + 2y 2 = 11 (e) (2, 3) w.r. to the circle x 2 + y 2 + 6x + 8y 96 = 0 (ii) Find the pole of the line: (a) x + 2y 1 = 0 w.r. to the circle x 2 + y 2 = 5 (b) 2x y + 16 = 0 w.r. to the circle x 2 + y 2 = 16 (c) x y + 2 = 0 w.r. to the circle x 2 + y 2 4x + 6y 12 = 0 (d) x 2y + 22 = 0 w.r. to the circle x 2 + y 2 5x + 8y + 6 = 0 (iii) Find the value of k if the points: (a) (1, 2), (k, 1) are conjugate, w.r. to x 2 + y 2 = 5 (b) (4, k), (2, k) are conjugate w.r. to x 2 + y 2 = 17 (c) (4, 2), (8, k) are conjugate w.r. to 3x 2 + 3y 2 12x + 4y 4 = 0 (iv) Find the value of k if the lines: (a) 2x + 3y 4 = 0, kx + 4y 2 = 0 are conjugate lines w.r. to the circle x 2 + y 2 = (i) Find the inverse point of (a) (1, 1) w.r. to the circle x 2 + y 2 = 4 (b) ( 2, 3) w.r. to the circle x 2 + y 2 4x 6y + 9 = Find the angle between the pair of tangents drawn from the point (a) (0, 0) to the circle x 2 + y 2 14x + 2y + 25 = 0 (b) (3, 2) to the circle x 2 + y 2 6x + 4y 2 = 0

9 (c) (4, 2) to the circle x 2 + y 2 = (i) Find the point of intersection of tangents drawn at the point where the circle: (a) x 2 + y 2 = 24 cuts the line 3x + 4y = 12 (b) x 2 + y 2 = 9 cuts the line 4x + 3y = 9 (ii) Show that the polar of the point (2t, t 4) w.r. to the circle x 2 + y 2 4x 6y + 1 = 0 always passes through the point (3, 1). 44. (i) For all real values of k, show that the polars of a given point with respect to the circle x 2 + y 2 2kx + c 2 = 0 pass through a fixed point. (ii) Prove that the polars of (1, 2) w.r. to circle x 2 + y 2 + 6y + 5 = 0 and x 2 + y 2 + 2x + 8y + 5 = 0 coincide. 2. (ii) a = 2, b = 6 4. (a) ( 3, 4) (b) (3, 2) 5. k = 8 6. (a) (x + 2) 2 + (y 3) 2 = 0 (b) (x 4) 2 + (y + 3) 2 = 0 (c) x 2 + y 2 = 0 7. (i) x 2 + y 2 + 4x 6y 13 = 0 (ii) x 2 + y 2 + 8x + 12y + 11 = (a) (7, 3) (b) ( 1, 5) 13. (ii) (a) 3x 2 + 3y 2 + 2x 2y 20 = 0 (b) 3x 2 + 3y 2 23x 13y + 4 = 0 (c) x 2 + y 2 ax by = (a) x 2 + y 2 4x 2y 5 = 0 (b) x 2 + y 2 2x + 2y 11 = x 2 + y 2 + 2ax + 2by = b 2 + q (a) 2 45 (b) 2 2 g C (c) (i) ( cos, cos ) (ii) (x 2) 2 + (y + 1) 2 = 4 2 (iii) (x 4) 2 + (y 3) 2 = x 2 + y 2 = a 2 + b x + y = (a) 3x + 2y 13 = 0 (b) 2x + 7y + 22 = 0 (c) 2x + y 4 = (a) k = 1 (b) k = (a) x 2 + y 2 + 4x 6y + 4 = 0 (b) x 2 + y 2 6x + 4y + 4 = 0 (d) x 2 + y 2 8x 6y + 21 = (i) (a) x + y = (b) 4x 3y = 24 and 4x 3y = 26 (ii) (a) (2, 3) (b) (5, 2)

10 26. 4 a x y (i) (3, 1) 29. (i) (a) 27 (b) 69 (c) 2a 2 + 2b 2 + 2a + 6b + 1= 0 (ii) (a) 4 (b) / (i) 6 (ii) k = 1 (iii) 5x 2 + 5y x + 7 = 0 (iv) (a) 21 (b) 9 (c) 2 3 (v) (a) k = 1 (b) k = (i) y = 2x (ii) 2y = 3x, ( 6, 9) (iii) ( 3, 2) 32. x 2 + y 2 8x 10y + 16 = 0 _ 33. (i) 1 24 tan 7 30/13 (ii) x 2 + y 2 = 2a 2 (iv) 3x 2 + 8xy 3y 2 40x 20y = 0 (v) x + y = 34. (i) k = 3 (ii) k = 3 (iii) (6, 8) 36. (i) 5) 2 2 a h a k, h k h k (ii) (10, 38. (i) x 2 + y 2 3x + y 4 = (a) 2x 3y = 5 (b) x + y = (i) (a) 3x + 4y = 25 (b) x = 0 (c) x + 2y = 7 (d) 6x + 2y = 11 (e) 5x + 7y = 78 (ii) (a) (5, 10) (b) ( 2, 1) (c) 11 4, 7 7 (iii) (a) k = 3 (b) k = 3 (c) k = 3 (iv) (a) k = (i) (a) (2, 2) (b) (1, 3) 42. (a) /2 (b) 43. (i) (a) (6, 8) (b) (4, 3) _ 1 15 tan 7 (c) /2

CIRCLES PART - II Theorem: The condition that the straight line lx + my + n = 0 may touch the circle x 2 + y 2 = a 2 is

CIRCLES PART - II Theorem: The equation of the tangent to the circle S = 0 at P(x 1, y 1 ) is S 1 = 0. Theorem: The equation of the normal to the circle S x + y + gx + fy + c = 0 at P(x 1, y 1 ) is (y