Question

The differential equation of the family of parabolas with vertex at $\left(0,1\right)$ and having axis along $\mathrm{y}$-axis is

• A. $\mathrm{yy}’+2\mathrm{xy}+1=0$
• B. $\mathrm{xy}’+\mathrm{y}+1=0$
• C. $\mathrm{xy}’–2\mathrm{y}–2=0$
• D. $\mathrm{xy}’–\mathrm{y}–1=0$

Answer 1

The correct option is C

$\mathrm{xy}’–2\mathrm{y}–2=0$

Explanation for the correct option:

Given : Vertex $(0,-1)$ and axis along $\mathrm{y}$-axis
Equation of family parabola having vertex at $(0,-1)$ and axis along $\mathrm{y}$-axis is
${\mathrm{x}}^2=4\mathrm{a}(\mathrm{y}+1)$
On differentiating w.r.t$\mathrm{x}$,
$$\begin{gathered} 2\mathrm{x}=4\mathrm{a}\frac{\mathrm{dy}}{\mathrm{dx}} \\ \mathrm{a}=\frac{\mathrm{x}}{2{\mathrm{y}}_1}\left(\because \frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{y}}_1\right) \end{gathered}$$
On substituting $a$ in equation ${\mathrm{x}}^2=4\mathrm{a}(\mathrm{y}+1)$ we get
${\mathrm{x}}^2=4\left(\frac{\mathrm{x}}{2{\mathrm{y}}_1}\right)(\mathrm{y}+1)$

$\Rightarrow {\mathrm{xy}}_1-2\mathrm{y}-2=0$
Hence, the correct answer is option (C)