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Mathematics

Mathematics. Session. Circle Sessions - 3. Session Objectives. Session Objective. Equation Of Family Of Circles Equation Of Chord Whose mid –Point is Given Equation Of Chord Of Contact Angle Between Two Circles Orthogonal Intersection Equation Of Common Chord.

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Mathematics

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  1. Mathematics

  2. Session Circle Sessions - 3

  3. Session Objectives

  4. Session Objective Equation Of Family Of Circles Equation Of Chord Whose mid –Point is Given Equation Of Chord Of Contact Angle Between Two Circles Orthogonal Intersection 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 = 0 L = 0 B A S+L = 0

  6. Questions

  7. Illustrative Problem Write the family of circles passing through the intersection of x2 + y2 –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 (x1,y1) and (x2,y2) B(x2,y2) A(x1,y1) S  (x-x1)(x-x2) + (y-y1)(y-y2) = 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 (x1,y1) S=0 A(x1,y1) L=0 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 x2+y2=25 at (3,4) and passes through (1,1). Solution: Tangent at (3,4) is 3x+4y-25=0 Therefore family of circle touching x2+y2-25 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(x1,y1) 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 S2 = 0 S1 = 0 S1 + S2 = 0 (  -1)

  21. Question

  22. Illustrative Problem Find the equation of circle passing through origin and the points of intersection of the two circles x2 + y2 - 4x - 6y – 3 = 0 and x2 + y2 + 4x - 2y – 4 = 0 Solution: Equation of family of circle is x2 + y2 - 4x - 6y – 3 + (x2 + y2 + 4x - 2y - 4) = 0 It passes through (0,0) =>x2 + y2 - 28x - 18y = 0

  23. Equation of chord whose mid-point is (x1,y1) S=0 C=(0,0) D A (x1,y1) B xx1 + yy1 = x12 + y12 T = S1

  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). Solution: Method 1 Equation of circle is

  26. Illustrative Problem C=(1,2) D A (2,1) B 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: Slope of AB = 1 Equation of AB is x – y – 1 = 0

  27. Equation of chord of contact of tangents drawn from a point (x1,y1) B(x3,y3) Equation of chord of contact is P(x1,y1) xx1+yy1=a2 or A(x2,y2) T = 0

  28. Question

  29. Illustrative problem Find the equation of chord of contact of a point (-2,-3) with respect to circle x2 + y2 - 2x - 6y + 1=0 Solution : Required circle is T = 0 x(-2) + y(-3) –2(-2) –6(-3) + 1 = 0 2x + 3y - 23 = 0

  30. Angle at which two circles intersect S1 =0 S2 = 0 P r1 r2 C1 C2  d = distance(c1,c2)

  31. Question

  32. Illustrative Problem C1 = (4,1); r1 = C2 = (-1,-4); r2 = Find the angle at which the circles x2 + y2 –8x – 2y - 9 =0 and x2 + y2 + 2x + 8y - 7 = 0 intersect. Solution :

  33. Orthogonal Intersection S1=0 S2=0 r1 r2 C2 C1 90 d = distance (c1,c2) Method 1: d2 = r12+r22

  34. Orthogonal Intersection S1=0 S2=0 r1 r2 C2 C1 90 Method 2: 2g1g2 + 2f1f2 = c1 + c2

  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. Solution : Two circles cut orthogonally Therefore a2 + a2 = 18 => a = 3

  37. Common Chord Of two circles S1 = 0 and S2 = 0 S1 =0 S2 =0 A B Equation of common chord is S1 - S2 = 0

  38. Question

  39. Illustrative Problem Find the equation of common chord of two circles x2 + y2 =25 and 4x2 + 4y2 - 40x + 91=0 Solution :

  40. Class Test

  41. A variable chord is drawn through theorigin to the circle x2 + y2 – 2ax = 0.The locus of the centre of the circledrawn on this chord as diameter is Class Exercise - 1

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

  43. 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 = a2 + b2 + k2 Class Exercise - 2

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

  45. Equation of the circle which passesthrough the origin, has its centre onthe line x + y = 4 and cuts the circlex2 + y2 – 4x + 2y + 4 = 0 orthogonally is (d) None of these Class Exercise - 3

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

  47. Class Exercise - 4 If O is the origin and OP, OQ aredistinct tangents to the circlex2 + y2 + 2gx +2fy + c = 0, thecircumcentre of the triangle OPQ is (a) (–g, –f) (b) (g, f)(c) (–f, –g) (d) None of these

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

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

  50. Class Exercise - 5 Prove that the circlex2 + y2 – 6x – 4y + 9 = 0 bisects thecircumference of the circlex2 + y2 – 8x – 6y + 23 = 0. Solution: Equation of common chord of the given circle isS1 – S2 = 0

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