Question

If $x = e ^{ t } \operatorname{cost}$ and $y = e ^{ t } \operatorname{sint}$, then what is $\frac{d x}{d y}$ at $t =0$ equal to?

• A. $-\frac{1}{2}$
• B. $-\frac{1}{4}$
• C. $0$
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

Given that, $x = e^tsint$, $y = e^tcost\,\ldots(i)$
At point $(1,1)$, $1 = e^t\, sin \,t$
and $1 = e^t\, cost$
$\Rightarrow tan\,t=1$
$\Rightarrow t=\frac{\pi}{4}$
On differentiating $\left(i\right)$ $w$.$r$.$t$. $t$, we get
$\frac{dy}{dt}=e^{t}\left(cos\,t-sin\,t\right)$
and $\frac{dx}{dt}=e^{t}\left(sin\,t+cos\,t\right)$
$\therefore \frac{dy}{dx}=\frac{dy/dt}{dx/dt}$
$=\frac{cos\,t-sin\,t}{cos\,t+sin\,t}$
$\frac{dy}{dx}\bigg|_{(1,1)}$ $= \frac{cos \frac{\pi}{4}-sin \frac{\pi}{4}}{cos \frac{\pi}{4}+sin \frac{\pi}{4}}$
$=\frac{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=0$