The parabolas C 1 : y 2 = 4 a x - a and C 2 : y 2 = - 4 a x - k intersect at two distinct points A and B . If the slope of the tangent at A on C 1 is same as the slope of the normal at B on C 2 , then the value of k is equal to

Question

The parabolas C 1 : y 2 = 4 a x - a and C 2 : y 2 = - 4 a x - k intersect at two distinct points A and B . If the slope of the tangent at A on C 1 is same as the slope of the normal at B on C 2 , then the value of k is equal to, 3 a, 2 a, a, 0

MARKS App · Accepted Answer

Let, A = p , q & B = p , - q Now q 2 = 4 a p - a & q 2 = - 4 a p - k …(i) ⇒ p - a = - p + k ⇒ p = a + k 2 …(ii) Now for C 1 , d y d x = 2 a y d y d x p , q = 2 a q & - d x d y p , - q = q 2 a ⇒ 2 a q = q 2 a ⇒ q 2 = 4 a 2 … (iii) Putting values from (ii) & (iii) in equation (i), we get, 4 a 2 = 4 a a + k 2 - a ⇒ a = k - a 2 ⇒ k = 3 a

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