Please solve RD Sharma class 12 chapter Higher Order Derivatives exercise 11.1 question 13 maths textbook solution

$\frac{-a}{y^{2}}$

HInt:

You must know about derivative of second order

Given:

\begin{aligned} &{\text { If } x=a(\theta+\sin \theta), y=a(1+\cos \theta)} \\ &\text { Prove that } \frac{d^{2} y}{d x^{2}}=\frac{-a}{y^{2}} \end{aligned}

Solution:

\begin{aligned} &{\text { Let } x=a(\theta+\sin \theta), y=a(1+\cos \theta)} \\ \end{aligned}

\begin{aligned} &\left.\frac{d x}{d \theta}=a[1+\cos \theta] \quad \text { ( } \frac{d}{d \theta} \theta=1, \frac{d}{d \theta} \sin \theta=\cos \theta\right) \\ &\frac{d y}{d \theta}=a(-\sin \theta) \\ &\frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{-\sin \theta}{(1+\cos \theta)} \end{aligned}

\begin{aligned} &\sin 2 \theta=2 \sin \theta \cos \theta \\ &\cos 2 \theta=2 \cos ^{2} \theta-1 \\ &\frac{d y}{d x}=\frac{-2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}} \end{aligned}

\begin{aligned} &\frac{d y}{d x}=-\tan \frac{\theta}{2} \\ &\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d \theta}\left(\frac{d y}{d x}\right) \frac{d \theta}{d x} \\ &\frac{d}{d \theta}\left(-\tan \frac{\theta}{2}\right) \cdot \frac{1}{a(1+\cos \theta)} \end{aligned}

\begin{aligned} &\frac{d^{2} y}{d x^{2}}=\frac{-1}{2} \sec ^{2} \frac{\theta}{2} \cdot \frac{1}{a(1+\cos \theta)} \\ &=\frac{-\sec ^{2} \frac{\theta}{2}}{2 a(1+\cos \theta)} \\ &\frac{-a}{y^{2}}=\frac{-a}{a(1+\cos \theta)^{2}}=\frac{-a}{a^{2}(1+\cos \theta)(1+\cos \theta)} \end{aligned}

\begin{aligned} &\frac{-1}{a\left(1+2 \cos ^{2} \frac{\theta}{2}-1\right)(1-\cos \theta)} \\ &=\frac{-1}{2 a \cos ^{2} \frac{\theta}{2}(1+\cos \theta)}=\frac{-\sec ^{2} \frac{\theta}{2}}{2 a(1+\cos \theta)} \end{aligned}

\begin{aligned} &R H L:-\frac{-a}{y^{2}}=\frac{-a}{a(1+\cos \theta)^{2}}=\frac{-1}{a\left(1+2 \cos ^{2} \frac{\theta}{2}-1\right)(1-\cos \theta)} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-\sec ^{2} \frac{\theta}{2}}{2 a(1+\cos \theta)}=\frac{-a}{y^{2}} \\ &\frac{d^{2} y}{d x^{2}}=\frac{-a}{y^{2}} \end{aligned}

Hence proved