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JEE Advanced2019MathematicsCircleActual

Let the circles C 1 : x 2 + y 2 = 9 and C 2 : x - 3 2 + y - 4 2 = 16 , intersect at the points X and Y . Suppose that another circle C 3 : x - h 2 + y - k 2 = r 2 satisfies the following conditions: i centre of C 3 is collinear with the centres of C 1 and C 2 i i C 1 and C 2 both lie inside C 3 , and i i i C 3 touches C 1 at M and C 2 at N . Let the line through X and Y intersect C 3 at Z and W , and let a common tan

Options

  1. AI V - S
  2. BI V - U
  3. CI I I - R
  4. DI - P

Correct answer

A. I V - S

Step-by-step solution

Given centre of C 1 , C 2 and C 3 are collinear hence 0 0 1 3 4 1 h k 1 = 0 3 k = 4 h …(i) ⇒ M N is diameter of C 3 M N = M C 1 + C 1 C 2 + C 2 N 2 r = r 1 + C 1 C 2 + r 2 ∵ M N = 2 r 2 r = 3 + 3 - 0 2 + 4 - 0 2 + 4 r = 6 …(ii) ⇒ Given C 3 touches C 1 at M So C 1 C 3 = r - 3 h 2 + k 2 = 9 …(iii) From (i) and (iii) h = ± 9 5 and k = ± 12 5 So centre of C 3 will be 9 5 , 12 5 ⇒ Now equation of common chord of C 1 and C 2 will be C 1 - C 2 = 0 6 x + 8 y = 18 Equation of line X Y is 3 x + 4 y = 9 …(iv) Distance of line

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