Q75 / 160 · AP EAMCET 2014 · Mathematics
The differential equation of the family of parabolas with vertex at $(0,-1)$ and having axis along the $Y$-axis is
- A$y y^{\prime}+2 x y+1=0$
- B$x y^{\prime}+y+1=0$
- Correct answerC$x y^{\prime}-2 y-2=0$
- D$x y^{\prime}-y-1=0$
Solution
Equation of family of parabola having vertex $(0,-1)$ and axis along $y$-axis is
$$
x^2=4 a(y+1)
$$
On differentiating both sides w.r.t. $x$, we get
$$
2 x=4 a y^{\prime} \Rightarrow a=\frac{x}{2 y^{\prime}}
$$
On putting $a=\frac{1}{2 y^{\prime}}$ in Eq. (i), we get
$$
\begin{array}{cc}
& x^2=4 \times \frac{x}{2 y^{\prime}}(y+1) \\
\Rightarrow & x=\frac{2}{y^{\prime}}(y+1) \\
\Rightarrow \quad & x y^{\prime}-2 y-2=0
\end{array}
$$
Hence, the correct answer is option: 'C'.