Question

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

Answer 1

$x=a(\theta -\sin \theta ),y=a(1+\cos \theta )$
$\Rightarrow \frac{dx}{d\theta }=a(1-\cos \theta )$
& $\frac{dx}{d\theta }=a[0+(-\sin \theta \left)\right]=-a\sin \theta$
$\therefore \frac{dy}{dx}=\frac{\left(\frac{dy}{d\theta }\right)}{\left(\frac{dx}{d\theta }\right)}=\frac{-a\sin \theta }{a(1-\cos \theta )}=\frac{-2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{2{\sin }^2\frac{\theta }{2}}=\frac{-\cos \frac{\theta }{2}}{\sin \frac{\theta }{2}}=-\cot \frac{\theta }{2}$