Q103 / 180 · COMEDK 2015 · Mathematics
The differential equation of the family of parabolas $y^{2}=4 a x$, where $a$ is parameter, is
- Correct answerA$\frac{d y}{d x}=\frac{y}{2 x}$
- B$\frac{d y}{d x}=-\frac{y}{2 x}$
- C$\frac{d y}{d x}=-\frac{2 y}{x}$
- D$\frac{d y}{d x}=\frac{2 y}{x}$
Solution
Given, equation of parabola is $y^{2}=4 a x \quad \text{...(i)}$
Differentiating $\mathrm{E} \mathrm{g}$. (i) w.r.t. $x$, we get
$$
\begin{aligned}
2 y \frac{d y}{d x} &=4 a \\
\Rightarrow \quad \frac{d y}{d x} &=\frac{2 a}{y} \Rightarrow a=\frac{y}{2} \frac{d y}{d x}
\end{aligned}
$$
Substituting the value of $a$ in Eq. (i), we get
$$
\begin{aligned}
y^{2} &=4 \cdot \frac{y}{2} \frac{d y}{d x} x \\
\Rightarrow \quad y^{2} &=2 x y \frac{d y}{d x} \Rightarrow \frac{d y}{d x}=\frac{y}{2 x}
\end{aligned}
$$
Hence, the correct answer is option: 'A'.