An ellipse with major axis (CA), 4 units along x axis and minor axis (DB), 2 units along y axis with origin as center is shown. What is the value of  $\left|\frac{d^{2}y}{d{x}^2}\right|$ at B?

a = 2, b = 1
 $$\begin{gathered} x=a\cos \theta ,y=b\sin \theta \\ x=2\cos \theta ,y=\sin \theta \Rightarrow \frac{dx}{d\theta }=-2\sin \theta ,\frac{dy}{d\theta }=\cos \theta \\ \frac{dy}{dx}=\frac{-\cos \theta }{2\sin \theta }=-\frac{1}{2}\cot \theta \Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{1}{2}\cos e{c}^2\theta \frac{d\theta }{dx} \end{gathered}$$
$\frac{1}{2}\cos e{c}^2\theta \frac{1}{-2\sin \theta }=-\frac{1}{4}{\sin }^3\theta \Rightarrow \left|\frac{d^{2}y}{d{x}^2}\right|$=  at  $B=\frac{1}{4{\sin }^3{90}^0}=\frac{1}{4}$