The differential equation of all parabolas whose axis is y-axis, is

The differential equation of all parabolas whose axis is y-axis, is
  1. $\frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$
  2. $\frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$
  3. $x \frac{d^2 y}{d x^2}-\frac{d y}{d x}=0$
  4. $\frac{d^2 y}{d x^2}-y=0$

Solution

$\begin{aligned} & \lim _{x \rightarrow 0} \frac{\sqrt{1-\cos x^2}}{1-\cos x} \\ & =\lim _{x \rightarrow 0} \frac{\sqrt{2 \sin ^2 \frac{x^2}{2}}}{2 \sin ^2 \frac{x}{2}}=\lim _{x \rightarrow 0} \frac{\sqrt{2} \sin \frac{x^2}{2}}{2 \sin ^2 \frac{x}{2}} \end{aligned}$ Dividing numerator and denominator by $\frac{x^2}{4}$, we get $\begin{aligned} & \frac{\sqrt{2} \operatorname{lt}_{x \rightarrow 0} \frac{\sin \frac{x^2}{x^2}}{\frac{x^2}{2} \times \frac{1}{2}}}{2 \operatorname{lt}_{x \rightarrow 0} \frac{\sin ^2 \frac{x}{2}}{\frac{x^2}{4}}} \\ & =\frac{2 \sqrt{2}}{2}=\sqrt{2}\end{aligned}$

Asked in: MHT CET 2021 (24 Sep Shift 2)

Practice more Hyperbola questions on Aicharya