Question

Find $\frac{dy}{dx}$ , if $x=a\left(\theta -\sin \theta \right),y=a\left(1+\cos \theta \right)$

Answer 1

We have,

$x=a(\theta -\sin \theta )$ ……. (1)

$y=a(1+\cos \theta )$ ……. (2)

On differentiating of equation $\left(1\right)$ w.r.t $\theta$, we get

$\frac{dx}{d\theta }=a(1-\cos \theta )$ …….. (3)

On differentiating of equation $\left(2\right)$ w.r.t $\theta$, we get

$\frac{dy}{d\theta }=a(0-\sin \theta )$

$\frac{dy}{d\theta }=-a\sin \theta$ …….. (4)

On dividing equation $\left(4\right)$ by equation $\left(3\right)$, we get

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

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

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

Hence, this is the answer.