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Tangent and normal are drawn at P 16 , 16 on the parabola y 2 = 16 x , which intersect the axis of the parabola at A &   B , respectively. If C is the center of the circle through the points P , A &   B and ∠ C P B = θ , then a value of tan ⁡ θ is:

Options

  1. A4 3
  2. B1 2
  3. C2
  4. D3

Correct answer

C. 2

Step-by-step solution

Equation of tangent at x 1 , y 1 is y y 1 = 2 a x + x 1 Equation of normal at x 1 , y 1 is y - y 1 = - y 1 2 a ( x - x 1 ) Here tangent and normal passes through P ( 16 , 16 ) ∴ Equation of tangent is 2 y = x + 16 Equation of normal is y + 2 x = 48 Tangent and normal intersect x-axis at A - 16 , 0 and B ( 24 , 0 ) Since, A P B is a right angled triangle, hence A B is the diameter for the circle. Also mid point of A B is center of the circle. Hence Centre = 4 , 0 Slope of P C = 4 3 Slope of P B = - 2 tan &#828

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