A normal with slope 6 1 is drawn from the point ( 0 , − α ) to the parabola x 2 = − 4 a y , where a 0 . Let L be the line passing through ( 0 , − α ) and parallel to the directrix of the parabol

Question

A normal with slope 6 ​ 1 ​ is drawn from the point ( 0 , − α ) to the parabola x 2 = − 4 a y , where a > 0 . Let L be the line passing through ( 0 , − α ) and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and B . Let r denote the length of the latus rectum and s denote the square of the length of the line segment A B . If r : s = 1 : 16 , then the value of 24ais ________.

TestHub · Accepted Answer

d x d y ​ = − 2 a x ​ ⇒ d x d y ​ ​ N ​ = − t Slope of normal = t 1 ​ = 6 ​ 1 ​ ⇒ t = 6 ​ Now, 2 at − at 2 + α ​ = t 1 ​ ⇒ − at 2 + α = 2 a ⇒ − 6 a + α = 2 a ⇒ α = 8 a For A and B ​ x 2 = − 4 a ( − 8 a ) ⇒ x 2 = 32 a 2 ⇒ x = ± 4 2 ​ a ∴ A ( − 4 2 ​ a , − 8 a ) , B ( 4 2 ​ a , − 8 a ) ∴ A B 2 = ( 8 2 ​ a ) 2 = 128 a 2 = s ​ ∴ Length of LR = r = 4 a ​ ⇒ s r ​ = 128 a 2 4 a ​ = 16 1 ​ ∴ 32 a = 16 ⇒ a = 2 1 ​ ∴ 24 a = 12 Ans. ​

Mathematics - Parabola Question with Solution | TestHub

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