Question

The differential equation of the family of parabolas with vertex at $(0,-1)$ and having axis along the $Y$ -axis is

• A. $y y^{'}+2 x y+1=0$
• B. $x y^{'}+y+1=0$
• C. $x y^{'}-2 y-2=0$
• D. $x y^{'}-y-1=0$

Answer 1

Answer: (C)

Equation of family of parabola having vertex $(0,-1)$ and axis along $y$ -axis is
$x^{2}=4 a(y+1)\,\,\,...(i)$
On differentiating both sides w.r.t. $x$, we get
$2 x=4 a y^{'}$
$ \Rightarrow a=\frac{x}{2 y^{'}}$
On putting $a=\frac{1}{2 y^{\prime}}$ in Eq. (i), we get
$x^{2}=4 \times \frac{x}{2 y^{'}}(y+1) $
$\Rightarrow x=\frac{2}{y^{'}}(y+1) $
$\Rightarrow x y^{'}-2 y-2=0$