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Let normals to the parabola y 2 = 4 x at variable points P t 1 2 , 2 t 1 and Q t 2 2 , 2 t 2 meet at the point R t 2 , 2 t , then the line joining P and Q always passes through a fixed point α , β , then the value of α + β is equal to

Correct answer

2

Step-by-step solution

t = - t 1 - 2 t 1 and t = - t 2 - 2 t 2 ⇒ t 1 , t 2 are the roots of the equation t = - x - 2 x ⇒ x 2 + t x + 2 = 0 has roots t 1   &   t 2 ⇒ t 1 + t 2 = - t and t 1 t 2 = 2 Now, the equation of P Q is t 1 + t 2 y = 2 x + 2 t 1 t 2 ⇒ - t y = 2 x + 4 ⇒ 2 x + 2 + t y = 0 ⇒ P Q always passes through the common point of x + 2 = 0 and y = 0 ⇒ α , β = - 2,0 ⇒ α β = 0 & α + β = - 2

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