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Tangents are drawn from a point P on the directrix of the parabola y^2 = 8x to touch the parabola at points A and B . If the radius of the circumcircle of PAB is 9 , then the area of PAB is:

Options

  1. A54
  2. B108
  3. C81
  4. D27

Correct answer

A. 54

Step-by-step solution

Given the parabola y^2 = 8x , we have 4a = 8 a = 2 . The point P lies on the directrix of the parabola. A standard property states that tangents drawn from any point on the directrix to a parabola are perpendicular to each other. Thus, APB = 90^ . In the right-angled triangle PAB , the circumcircle has the hypotenuse AB as its diameter. Given the radius of the circumcircle is 9 , the length of the focal chord AB is 2 9 = 18 . The length of a focal chord in terms of parameter t is given by l = a (t + 1 t )^2 . Subst

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