Question

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

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

Answer 1

Answer: (C)

Equation of family of parabola having vertex $(0,-1)$ and axis along $y$ -axis is
$x^2=4a(y+1)...(i)$
On differentiating both sides w.r.t. $x$, we get
$2x=4a{y}^{{}^{\prime }}$
$\Rightarrow a=\frac{x}{2y^{{}^{\prime }}}$
On putting $a=\frac{1}{2y^{\prime }}$ in Eq. (i), we get
$x^2=4\times \frac{x}{2y^{{}^{\prime }}}(y+1)$
$\Rightarrow x=\frac{2}{y^{{}^{\prime }}}(y+1)$
$\Rightarrow x{y}^{{}^{\prime }}-2y-2=0$