# Prove the following?

## If $x = a \sin \theta$ and $y = b \sin \theta$, Prove that ${a}^{2} / {x}^{2} - {b}^{2} / {y}^{2} = 1$

Apr 4, 2018

color(red)(y=bsintheta
I think
color(red)(y=btantheta is O.K.

#### Explanation:

We have ,

color(red)(x=asintheta and y=bsintheta

$\implies \frac{a}{x} = \frac{1}{\sin} \theta \mathmr{and} \frac{b}{y} = \frac{1}{\sin} \theta$

Squaring both sides of both equn .

${a}^{2} / {x}^{2} = {\csc}^{2} \theta \mathmr{and} {b}^{2} / {y}^{2} = {\csc}^{2} \theta$

Comparing both equn. for ${\csc}^{2} x$

${a}^{2} / {x}^{2} = {b}^{2} / {y}^{2}$

${a}^{2} / {x}^{2} - {b}^{2} / {y}^{2} = 0$

So ,

${a}^{2} / {x}^{2} - {b}^{2} / {y}^{2} \ne 1$

If we take,

color(red)(x=asintheta and y=btantheta

$\implies \frac{a}{x} = \frac{1}{\sin} \theta = \csc \theta \mathmr{and} \frac{b}{y} = \frac{1}{\tan} \theta = \cot \theta$

Squaring both sides of both equn.

${a}^{2} / {x}^{2} = {\csc}^{2} \theta \ldots \to \left(I\right)$

${b}^{2} / {y}^{2} = {\cot}^{2} \theta \ldots \to \left(I I\right)$

We know that,

${\csc}^{2} \theta - {\cot}^{2} \theta = 1$

Using $\left(I\right) \mathmr{and} \left(I I\right)$ ,we get

${a}^{2} / {x}^{2} - {b}^{2} / {y}^{2} = 1$