Question

A curve is represented parametrically by the equations $x=e\cos t$ and $y=e\sin t$ where $t$ is a parameter. Then

The value of $\frac{d^{2}y}{d{x}^2}$ at the point where $t=0$ is

Answer 1

Given:

$x=ecost$ and $y=esint$

Hence,

$cost=\frac{x}{e}$ and.........[1]

$sint=\frac{y}{e}$..................[2]

Squaring and adding both,

$x^2+y^2=e^2$.......................[3]

Differentiating equation [3],

$2x+2y\left(dy/dx\right)=0$.................[4]

$⟹\frac{dy}{dx}=-\frac{x}{y}$

Again differentiating [4]:

$2+2y\left(d^{2}y/d{x}^2\right)+2(dy/dx{)}^2=0$

$⟹\frac{d^{2}y}{d{x}^2}=\frac{-1-(-x/y{)}^2}{y}=\frac{-y^2-x^2}{y^3}=\frac{-(esint{)}^2-(ecost{)}^2}{(esint{)}^3}$

Now, at $t=0$

$\frac{d^{2}y}{d{x}^2}=\frac{0-e^2}{0}=-\infty$