Question

If $x$ and $y$ are connected parametrically by the equation given exercise without eliminating the parameter, Find $\frac{dy}{dx}$.
$x=a(\theta -\sin \theta ),y=a(1+\cos \theta )$

Answer 1

Here, $x=a(\theta -\sin \theta )$

$y=a(1+\cos \theta )$

Differentiating both sides with respect to x, we have

$\frac{dx}{d\theta }=\frac{d}{d\theta }\left[a\right(\theta -\sin \theta \left)\right]$

$=a(1-\cos \theta )$

$\frac{dy}{d\theta }=\frac{d}{d\theta }(1+\cos \theta )=-a\sin \theta$

$\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}=\frac{-a\sin \theta }{a(1-\cos \theta )}$

$=-\frac{2\cos \frac{\theta }{2}\sin \frac{\theta }{2}}{2{\sin }^2\frac{\theta }{2}}$

$=-\cot \frac{\theta }{2}$