1. Mathematics

2. Circle Sessions - 3 Session

3. Session Objectives

4. Session Objective 1.Equation Of Family Of Circles 2.Equation Of Chord Whose mid –Point is Given 3.Equation Of Chord Of Contact 4.Angle Between Two Circles 5.Orthogonal Intersection 6.Equation Of Common Chord

5. Equation Of Family Of Circles Family of circles passing through the intersection of a circle S = 0 and a line L = 0 is S = 0L = 0 A B S+L = 0

6. Questions

7. Illustrative Problem Write the family of circles passing through the intersection of x 2 + y 2 –9 = 0 and x + y –1 = 0. Find that member of this family which passes through the origin. Solution: Family of required circle is S + L = 0

8. Solution Cond. Since the required circle passes through the origin, we have (0+0-9) + (0+0-1) = 0 = -9 Substituting value of in (1) we get

9. Equation Of Family Of Circles Passing through points (x 1,y 1 ) and (x 2,y 2 ) A(x 1,y 1 ) B(x 2,y 2 ) S  (x-x 1 )(x-x 2 ) + (y-y 1 )(y-y 2 ) = 0 Family of circles is S + L = 0

10. Questions

11. Illustrative Problem Find the equation of the circle passing through the (4,1), (6,5) and having its centre on 4x+3y-16=0 Solution: Circle whose end points of diameter are (4,1), (6,5) is Equation of line passing through (4,1) and (6,5) is 2x-y-7=0. Therefore family of circles is

12. Solution Cond.

13. Equation of family of circle which touches a given circle S at a given point (x 1,y 1 ) S=0 L=0 A(x 1,y 1 ) S + L = 0 where L = 0 is tangent at the given point on it.

14. Questions

15. Illustrative problem Find the equation of the circle which touches the circle x 2 +y 2 =25 at (3,4) and passes through (1,1). Solution: Tangent at (3,4) is 3x+4y-25=0 Therefore family of circle touching x 2 +y 2 -25 at (3,4) is

16. Solution Cond. It passes thorough (1,1) 1 + 1 – 25 + (3x+4y-25) = 0 Therefore the required equation of the circle is

17. Equation of circles which touches a given line at a given point on it A(x 1,y 1 ) L = 0 family of circle is S + L=0

18. Questions

19. Illustrative Problem Find the equation of the circle passing through origin(0,0) and touching the line 2x+y-1=0 at (1,-1) Solution: Family of the circles touching 2x+y-1 = 0 at (1,-1) is It passes through (0,0)  = 2

20. Equation of Circle Passing through the points of intersection of two circles S 1 + S 2 = 0 (  -1) S 1 = 0 S 2 = 0

21. Question

22. Illustrative Problem Find the equation of circle passing through origin and the points of intersection of the two circles x 2 + y 2 - 4x - 6y – 3 = 0 and x 2 + y 2 + 4x - 2y – 4 = 0 Solution: Equation of family of circle is x 2 + y 2 - 4x - 6y – 3 + (x 2 + y 2 + 4x - 2y - 4) = 0 It passes through (0,0) =>x 2 + y 2 - 28x - 18y = 0

23. Equation of chord whose mid-point is (x 1,y 1 ) S=0 C=(0,0) AB(x 1,y 1 ) D xx 1 + yy 1 = x 1 2 + y 1 2 T = S 1

24. Questions

25. Illustrative Problem Find the equation of the chord of the circle (x-1) 2 + (y-2) 2 = 4 whose mid – point is (2,1). Equation of circle is Solution: Method 1

26. Illustrative Problem Find the equation of the chord of the circle (x-1) 2 + (y-2) 2 = 4 whose mid – point is (2,1). Solution Method 2: C=(1,2) AB(2,1) D Slope of AB = 1 Equation of AB is x – y – 1 = 0

27. Equation of chord of contact of tangents drawn from a point (x 1,y 1 ) P(x 1,y 1 ) A(x 2,y 2 ) B(x 3,y 3 ) Equation of chord of contact is xx 1 +yy 1 =a 2 or T = 0

28. Question

29. Illustrative problem Find the equation of chord of contact of a point (-2,-3) with respect to circle x 2 + y 2 - 2x - 6y + 1=0 Required circle is T = 0 x(-2) + y(-3) –2(-2) –6(-3) + 1 = 0 2x + 3y - 23 = 0 Solution :

30. Angle at which two circles intersect S 1 =0S 2 = 0 C1C1 C2C2 P  r1r1 r2r2 d = distance(c 1,c 2 )

31. Question

32. Illustrative Problem Find the angle at which the circles x 2 + y 2 –8x – 2y - 9 =0 and x 2 + y 2 + 2x + 8y - 7 = 0 intersect. C 1 = (4,1); r 1 = C 2 = (-1,-4); r 2 = Solution :

33. Orthogonal Intersection C1C1 C2C2 r1r1 r2r2 S 1 =0S 2 =0 90 d = distance (c 1,c 2 ) Method 1: d 2 = r 1 2 +r 2 2

34. Orthogonal Intersection C1C1 C2C2 r1r1 r2r2 S 1 =0S 2 =0 90 Method 2: 2g 1 g 2 + 2f 1 f 2 = c 1 + c 2

35. Question

36. Illustrative Problem If two circles of equal radii ‘a’ with centre (2,3) and (5,6) respectively cut each other orthogonally then find the value of a. Two circles cut orthogonally Therefore a 2 + a 2 = 18 => a = 3 Solution :

37. Common Chord Of two circles S 1 = 0 and S 2 = 0 S 1 =0 S 2 =0 A B Equation of common chord is S 1 - S 2 = 0

38. Question

39. Illustrative Problem Find the equation of common chord of two circles x 2 + y 2 =25 and 4x 2 + 4y 2 - 40x + 91=0 Solution :

40. Class Test

41. Class Exercise - 1 A variable chord is drawn through the origin to the circle x 2 + y 2 – 2ax = 0. The locus of the centre of the circle drawn on this chord as diameter is

42. Solution Let (h, k) be the centre of the circle. Then (h, k) being the mid-point of the chord of the given circle Since it passes through (0, 0) Locus is Hence, answer is (c).

43. Class Exercise - 2 If the circle passes through the point (a, b) and cuts the circle orthogonally, equation of the locus of its centre is (a) 2ax + 2by = a 2 + b 2 + k 2

44. Solution Let (h, k) be the centre  Locus is 2ax + 2by – Hence, answer is (a).

45. Class Exercise - 3 Equation of the circle which passes through the origin, has its centre on the line x + y = 4 and cuts the circle x 2 + y 2 – 4x + 2y + 4 = 0 orthogonally is (d) None of these

46. Solution Let centre of the circle is (g, 4 – g) [its centre is on x + y = 4] Equation of circle is [ it passes through origin] Since it cuts the given circle orthogonally, 2 × g × 2 – 2 (4 – g) = 4 6g = 12  g = 2  Equation of the required circle is Hence, answer is (c).

47. Class Exercise - 4 If O is the origin and OP, OQ are distinct tangents to the circle x 2 + y 2 + 2gx +2fy + c = 0, the circumcentre of the triangle OPQ is (a) (–g, –f)(b) (g, f) (c) (–f, –g)(d) None of these

48. Solution PQ is chord of contact of (0, 0)  Equation of PQ is gx + fy + c = 0 Family of circles passing through PQ and given circle is

49. Solution Cont. It passes through (0, 0) Hence, answer is (d).

50. Class Exercise - 5 Prove that the circle x 2 + y 2 – 6x – 4y + 9 = 0 bisects the circumference of the circle x 2 + y 2 – 8x – 6y + 23 = 0. Solution: Equation of common chord of the given circle is S 1 – S 2 = 0

51. Solution Contd.. Centre of circle is (4, 3) which lies on x + y – 7 = 0, first circle bisects the circumference of the second circle because common chord passes through the end points of a diameter of the second circle.

52. Class Exercise - 6 If OA and OB are two equal chords of the circle x 2 + y 2 – 2x + 4y = 0 perpendicular to each other and passing through the origin then find the equation of OA and OB. Solution: Let chords be y = mx and. Since chords are of equal lengths, perpendicular distance from the centre to the chords will be same.

53. Solution Contd.. Equations of OA and OB are x + 3y = 0 and 3x – y = 0

54. Class Exercise - 7 The coordinate of two points P and Q are (2, 3) and (3, 2) respectively. If circles are described on OP and OQ as diameters, O being the origin then find the length of their common chord. Solution: Therefore OR is perpendicular to PQ. We have to find the length of OR. From figure it is clear that OR is length from origin to the line PQ.

55. Solution Contd.. Equation of PQ is y – 2 = –1 (x – 3)

56. Class Exercise - 8 Determine the equation of the circle whose diameter is the chord x + y = 1 of the circle x 2 + y 2 = 4. Solution: Equation of family of circles passing through the intersection of circle and line is

57. Solution Contd.. Its centre lies on x + y – 1 = 0 Substituting value of in the equation (i), we get

58. Class Exercise - 9 Consider a family of circles passing through two fixed points A (3, 7)and B (6, 5). Show that the chords in which the circle x 2 + y 2 – 4x – 6y – 3 = 0 cuts the members of the family are concurrent at a point. Find the coordinate of this point. Solution: Family of circles is (x – 3) (x – 6) + (y – 7) (y – 5)

59. Solution Contd.. Common chord is S 1 – S 2 = 0 This chord is intersection of –5x – 6y + 56 = 0 and 2x + 3y – 27 = 0 Solving these equations we get.

60. Class Exercise - 10 If two chords, drawn from the point (p, q) on the circle x2 + y2 = px + qy (where pr 0) are bisected by the x-axis, then (a) p 2 = q 2 (b) p 2 = 8q 2 (c) p 2 8q 2 Solution: Let the chord is bisected at A (h, 0) P is (p, q)

61. Solution Contd.. Q (2h – p, –q) lies on circle This is quadratic in h and h is real and distinct

62. Thank you