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JEE Main202126 Aug 2021Evening ShiftMathematicsCircleActual

A circle C touches the line x = 2 y at the point 2 , 1 and intersects the circle C 1 : x 2 + y 2 + 2 y - 5 = 0 at two points P and Q such that P Q is a diameter of C 1 . Then the diameter of C is :

Options

  1. A4 15
  2. B285
  3. C15
  4. D7 5

Correct answer

D. 7 5

Step-by-step solution

Family of circle touching line 2 y = x at point 2 , 1 ( x - 2 ) 2 + ( y - 1 ) 2 + λ ( x - 2 y ) = 0       . . . ( 1 ) common chord P Q is ( x - 2 ) 2 + ( y - 1 ) 2 + λ ( x - 2 y ) - x 2 - y 2 - 2 y + 5 = 0 (diameter of C 1 passes through ( 0 , - 1 ) ) 4 + 4 + 2 λ - 1 + 2 + 5 = 0 λ = - 7 put in equation 1 ( x - 2 ) 2 + ( y - 1 ) 2 - 7 ( x - 2 y ) = 0 x 2 + y 2 - 11 x + 12 y + 5 = 0 r = 121 4 + 36 - 5 = 121 + 124 4 = 245 4 = 245 2 ⇒  diameter  = 245 = 7 5

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