# Write the correct alternative in the following:

Question:

Write the correct alternative in the following:

If $x=f(t) \cos t-f^{\prime}(t) \sin t$ and $y=f(t) \sin t+f^{\prime}(t) \cos t, t h e n\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}=$

A. $f(t)-f^{\prime \prime}(t)$

B. $\left\{f(t)-f^{\prime \prime}(t)\right\}^{2}$

C. $\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$

D. none of these

Solution:

Given:

$x=f(t) \cos t-f^{\prime}(t) \sin t$

$y=f(t) \sin t+f^{\prime}(t) \cos t$

$\frac{d x}{d t}=f^{\prime}(t) \cos t-f(t) \sin t-f^{\prime}(t) \sin t-f^{\prime}(t) \cos t$

$=-f(t) \sin t-f^{\prime \prime}(t) \sin t$

$=-\sin t\left[f(t)+f^{\prime \prime}(t)\right]$

$\left(\frac{d x}{d t}\right)^{2}=\left\{-\sin t\left[f(t)+f^{\prime}(t)\right]\right\}^{2}$

$=(\sin t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$

$\frac{d y}{d t}=f^{\prime}(t) \sin t+f(t) \cos t+f^{\prime \prime}(t) \cos t-f^{\prime}(t) \sin t$

$=f(t) \cos t+f^{\prime \prime}(t) \cos t$

$=\cos t\left[f(t)+f^{\prime \prime}(t)\right]$

$\left(\frac{d y}{d t}\right)^{2}=\left\{\cos t\left[f(t)+f^{\prime \prime}(t)\right]\right\}^{2}$

$=(\cos t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$

$\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}=(\sin t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}+(\cos t)^{2}\left\{f(t)+f^{\prime \prime}(t)\right\}^{2}$

$=\left\{\mathrm{f}(\mathrm{t})+\mathrm{f}^{\prime \prime}(\mathrm{t})\right\}^{2}$