Question
If $\text{x}=\text{a}\sin\theta\text{ and y}=\text{b}\cos\theta,$ what is the value pf $b^2x^2 + a^2y^2?$
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Answer
$\text{x}=\text{a}\sin\theta,\text{y}=\text{b}\cos\theta$
$\frac{\text{x}}{\text{a}}=\sin\theta,\frac{\text{y}}{\text{b}}=\cos\theta$
Squaring and adding we get,
$\therefore\ \frac{\text{x}^2}{\text{a}^2}=\sin^2\theta,\frac{\text{y}^2}{\text{b}^2}=\cos^2\theta$
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=\sin^2\theta+\cos^2\theta$
$\Rightarrow\ \frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$
$\Rightarrow\ \frac{\text{b}^2\text{x}^2+\text{a}^2\text{y}^2}{\text{a}^2\text{b}^2}=1\Rightarrow\ \text{b}^2\text{x}^2+\text{a}^2\text{y}^2=\text{a}^2\text{b}^2$
$\therefore\ \text{b}^2\text{x}^2+\text{a}^2\text{y}^2=\text{a}^2\text{b}^2$
$\frac{\text{x}}{\text{a}}=\sin\theta,\frac{\text{y}}{\text{b}}=\cos\theta$
Squaring and adding we get,
$\therefore\ \frac{\text{x}^2}{\text{a}^2}=\sin^2\theta,\frac{\text{y}^2}{\text{b}^2}=\cos^2\theta$
$\frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=\sin^2\theta+\cos^2\theta$
$\Rightarrow\ \frac{\text{x}^2}{\text{a}^2}+\frac{\text{y}^2}{\text{b}^2}=1$
$\Rightarrow\ \frac{\text{b}^2\text{x}^2+\text{a}^2\text{y}^2}{\text{a}^2\text{b}^2}=1\Rightarrow\ \text{b}^2\text{x}^2+\text{a}^2\text{y}^2=\text{a}^2\text{b}^2$
$\therefore\ \text{b}^2\text{x}^2+\text{a}^2\text{y}^2=\text{a}^2\text{b}^2$
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