The two curves `x=y^2,x y=a^3` cut orthogonally at a point. Then `a^2` is equal to `1/3` (b) 3 (c) 2 (d) `1/2`

Options

A. `(1)/(3)`
B. 3
C. 2
D. `(1)/(2)`

Answer

D

Explanation

We have,
`x=y^(2) " " (i) and , xy =a^(3) " " …(ii) `
These two curves intersect at `P(a, a^(2))`.
Now,
`x=y^(2) rArr (dy)/(dx)=(1)/(2y) rArr m_(1) = ((dy)/(dx))_(p) = (1)/(2a^(2))`
and,
`xy=a rArr x(dy)/(dx)+y=0 rArr (dy)/(dx) = (y)/(x) rArr m_(2) = ((dy)/(dx))_(p) = -a`
If the two curves intersect orthogonally, then
`m_(1)m_(2) = -1 rArr (1)/(2a^(2)) xx -a = -1 rArr a = (1)/(2)`