First slide

Hyperbola in conic sections

Question

A and B are two points on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $O$ is the centre. If $O A$ is perpendicular to $O B$ then $\frac{1}{(O A)^{2}} + \frac{1}{(O B)^{2}}$ is equal to

Moderate

A

$\frac{1}{a^{2}} + \frac{1}{b^{2}}$
B

$\frac{1}{a^{2}} - \frac{1}{b^{2}}$
C

$\frac{1}{b^{2}} - \frac{1}{a^{2}}$
D

$a^{2} + b^{2}$

Solution

Let $O A = r_{1}$ and the coordinates of A be $(r_{1} \cos α, r_{1} \sin α)$
$(r_{2} \cos (α + \frac{π}{2}), r_{2} \sin (α + \frac{π}{2}))$
As A, B lie on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
$r_{1}^{2} (\frac{\cos^{2} α}{a^{2}} - \frac{\sin^{2} α}{b^{2}}) = 1$
$r_{2}^{2} (\frac{\sin^{2} α}{a^{2}} - \frac{\cos^{2} α}{b^{2}}) = 1 \Rightarrow \frac{1}{(O A)^{2}} + \frac{1}{(O B)^{2}}$
$= \frac{1}{r_{1}^{2}} + \frac{1}{r_{2}^{2}} = \frac{1}{a^{2}} - \frac{1}{b^{2}}$ .

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