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A tangent and a normal are drawn at the point P 8,8 on the parabola y 2 = 8 x which cuts the axis of the parabola at the points A and B respectively. If the centre of the circle through P , A and B is C , then the sum of s i n ∠ P C B and c o t ∠ P C B is equal to

Correct answer

1.55

Step-by-step solution

Let, 2 t 2 , 4 t = 8,8 = P ⇒ t = 2 Equation of the tangent at 8,8 is 2 y = x + 8 Equation of the normal at 8,8 is y = - 2 x + 24 Tangent & normal at a t 2 , 2 a t are t y = x + a t 2 & y = - t x + 2 a t + a t 3 ⇒ A = - 8 ,0 & B = 12 ,0 Because Δ P A B is a right-angled triangle Hence, the centre of the circumcircle is the mid-point of the hypotenuse A B is C = 2 ,0 The slope of line P C is 8 - 0 8 - 2 = 4 3 ⇒ t a n ∠ P C B = 4 3 , s i n ∠ P C B = 4 5 ⇒ c o t ∠ P C B + s i n ∠ P C B = 0 . 75 + 0 . 8 = 1.5 5

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