Let l x - 2 y = 1 intersects the parabola y 2 = 4 a x at points P and Q . If P S and Q S meet the parabola again at R and T respectively (where S is the focus of y 2 = 4 a x ), then the slope of R T is equal to

Question

Let l x - 2 y = 1 intersects the parabola y 2 = 4 a x at points P and Q . If P S and Q S meet the parabola again at R and T respectively (where S is the focus of y 2 = 4 a x ), then the slope of R T is equal to,

MARKS App · Accepted Answer

Let, P = t 1 2 , 2 t 1 and Q = t 2 2 , 2 t 2 Equation of P S is t 1 + t 2 y = 2 x + 2 t 1 t 2 ⇒ lines - 2 x + t 1 + t 2 y = 2 t 1 t 2 and l x - 2 y = 1 are same ⇒ - 2 l = t 1 + t 2 - 2 = 2 t 1 t 2 1 ⇒ t 1 + t 2 = 4 l & t 1 t 2 = - 1 l Now, R = 1 t 1 2 , - 2 t 1 and T = 1 t 2 2 , - 2 t 2 The slope of R T is - 2 t 2 + 2 t 1 1 t 2 2 - 1 t 1 2 = - 2 t 1 - t 2 t 1 t 2 t 1 - t 2 t 1 + t 2 t 1 2 t 2 2 = - 2 t 1 t 2 t 1 + t 2 = 2 l 4 l = 1 2 = 0 . 5

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