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JEE MainMathematicsCircle

A point P moves such that the sum of the squares of its distances from the points A(1, -3) and B(-3, -7) is k . Let the locus of P intersect the x -axis at two points and the y -axis at two points, forming a quadrilateral. If the area of this quadrilateral is 70 sq. units, then the value of k is

Options

  1. A92
  2. B20
  3. C116
  4. D58

Correct answer

C. 116

Step-by-step solution

Let P(x, y) be the moving point. Given, PA^2 + PB^2 = k (x-1)^2 + (y+3)^2 + (x+3)^2 + (y+7)^2 = k x^2 - 2x + 1 + y^2 + 6y + 9 + x^2 + 6x + 9 + y^2 + 14y + 49 = k 2x^2 + 2y^2 + 4x + 20y + 68 - k = 0 x^2 + y^2 + 2x + 10y + 68-k 2 = 0 This is the equation of a circle with g = 1 , f = 5 , and c = 68-k 2 . Length of the x -intercept = 2 g^2 - c = 2 1 - 68-k 2 = 2k - 132 Length of the y -intercept = 2 f^2 - c = 2 25 - 68-k 2 = 2k - 36 The quadrilateral is formed by the intercepts on the coordinate axes. Since the axes ar

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