If A & B are points on the parabola y2 = 4ax with vertex O such that OA perpendicular to OB & having lengths r1 & r2 respectively, then the value of \frac{r_{1}^{4 / 3} r_{2}^{4 / 3}}{r_{1}^{2 / 3} + r

Engineering

Mathematics

Chord of Parabola

Question

If A & B are points on the parabola y² = 4ax with vertex O such that OA perpendicular to OB & having lengths r₁ & r₂ respectively, then the value of $\frac{r_{1}^{4 / 3} r_{2}^{4 / 3}}{r_{1}^{2 / 3} + r_{2}^{2 / 3}}$ is :

none of these

a²

16a²

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r₁² sin² θ = 4 ar₁ cos θ — (1)
and r₂² cos² θ = 4 ar₂ sin θ — (2)
from (1) r₁² sin⁴ θ = 16 a² cos² θ — (3)
(3) $\div$ (1)
$\frac{r_{1}^{2} \sin^{3} θ}{4 a r_{2}}$ = $\frac{16 a^{2}}{r_{2}^{2}}$ ⟹ sin³ θ = $\frac{64 a^{3}}{r_{1}^{2} r_{2}}$ — (4)

Similarly cos³ θ = $\frac{64 a^{3}}{r_{1} r_{2}^{2}}$ — (5)