A and B are two points on the parabola `y^(2) = 4ax` with vertex O. if OA is perpendicular to OB and they have lengths `r_(1)` and `r_(2)` respectively, then the valye of `(r_(1)^(4//3)r_(2)^(4//3))/(r_(1)^(2//3)+r_(2)^(2//3))` is

Options

A. `16a^(2)`
B. `a^(2)`
C. `4a`
D. None of these

Answer

A

Explanation

`r_(1)^(2) sin^(2) theta = 4ar_(1) cos theta` (i)
and `r_(2)^(2) cos^(2) theta = 4ar_(2) sin theta` (ii)
From Eq. (i), we get `r_(1)^(2) sin^(4) theta = 16a^(2) cos^(2) theta` (iii)
from Eqs. (ii) and (iii), we get
`sin^(3) theta = (64a^(3))/(r_(1)^(2)r_(2))` (iv)
similarly, `cos^(3) theta = (64a^(3))/(r_(1)r_(2)^(2))`
Eliminating `theta`, we get
`(r_(1)^(4//3)r_(2)^(4//3))/(r_(1)^(2//3)+r_(2)^(2//3)) =16a^(2)`