Question

If $x=e^t\mathrm{cost}$ and $y=e^t\mathrm{sint}$, then what is $\frac{dx}{dy}$ 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, $x=e^{t}sint$, $y=e^{t}cost\dots (i)$
At point $(1,1)$, $1=e^{t}sint$
and $1=e^{t}cost$
$\Rightarrow tant=1$
$\Rightarrow t=\frac{\pi }{4}$
On differentiating $\left(i\right)$ $w$.$r$.$t$. $t$, we get
$\frac{dy}{dt}=e^t\left(cost-sint\right)$
and $\frac{dx}{dt}=e^t\left(sint+cost\right)$
$\therefore \frac{dy}{dx}=\frac{dy/dt}{dx/dt}$
$=\frac{cost-sint}{cost+sint}$
$\frac{dy}{dx}{|}_{(1,1)}$ $=\frac{cos\frac{\pi }{4}-sin\frac{\pi }{4}}{cos\frac{\pi }{4}+sin\frac{\pi }{4}}$
$=\frac{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}=0$