Question
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{a}(\theta+\sin\theta)$ and $\text{y}=\text{a}(1-\cos\theta)$
$\text{x}=\text{a}(\theta+\sin\theta)$ and $\text{y}=\text{a}(1-\cos\theta)$
Differentiating it with respect to $\theta$
$\frac{\text{dx}}{\text{d}\theta}=\text{a}(1+\cos\theta).....(\text{i}) $
And, $\text{y}=\text{a}(1-\cos\theta)$
Differentiating it with respect to $\theta$,
$ \frac{\text{dy}}{\text{d}\theta}=\text{a}(\theta+\sin\theta)$
and
$\frac{\text{dy}}{\text{d}\theta}=\text{a}\sin\theta...(\text{ii}) $
Using equation (i) and (ii),
$=\frac{\text{a}\sin\theta}{\text{a}(1-\cos\theta)} $
$=\frac{\frac{2\sin\theta}{2}\frac{\cos\theta}{2}}{\frac{2\sin^{2}\theta}{2}}, \begin{Bmatrix} \text{Since, }1-\cos\theta=\frac{2\sin^{2\theta}}{2}\\\frac{2\sin\theta}{2}\frac{\cos\theta}{2}=\sin\theta \end{Bmatrix}$
$=\frac{\text{dy}}{\text{dx}}=\frac{\tan}{2}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
(1) (0, 0, 0) and (1, 0, 2)
(2) (1, 3, 0) and (0, 3, 0)
$\frac{\text{dy}}{\text{dx}}-\frac{\text{y}}{\text{x}}+\text{cosec}\Big(\frac{\text{y}}{\text{x}}\Big)=0;\ \text{y}=0\ \text{when x}=1$