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Let PQ be a focal chord of the parabola y^2 = 16x , where S is the focus of the parabola. If a circle drawn with SQ as its diameter touches the y -axis at the point A(0, -8) , then the length of the focal chord PQ is equal to :

Options

  1. A25
  2. B20
  3. C64
  4. D10

Correct answer

A. 25

Step-by-step solution

For the parabola y^2 = 16x , we have a = 4 . The focus is S(4, 0) . Let the coordinates of Q be (4t'^2, 8t') . The equation of the circle with SQ as diameter is: (x - 4)(x - 4t'^2) + (y - 0)(y - 8t') = 0 Since the circle touches the y -axis, we substitute x = 0 to find the point of contact: (-4)(-4t'^2) + y^2 - 8t'y = 0 y^2 - 8t'y + 16t'^2 = 0 (y - 4t')^2 = 0 y = 4t' The circle touches the y -axis at (0, 4t') . It is given that this point is A(0, -8) . 4t' = -8 t' = -2 Since PQ is a focal chord, the parameter t of

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