Q94 / 180 · KCET 2009 · Mathematics
The differential equation of the family of circles passing through the orign and having their centres on the $x$-axis is
- Correct answerA$\mathrm{y}^{2}=\mathrm{x}^{2}+2 \mathrm{xy} \frac{\mathrm{dy}}{\mathrm{dx}}$
- B$y^{2}=x^{2}-2 x y \frac{d y}{d x}$
- C$x^{2}=y^{2}+x y \frac{d y}{d x}$
- D$x^{2}=y^{2}+3 x y \frac{d y}{d x}$
Solution
The system of circles passing through origin and centre lies on $\mathrm{x}$-axis is
$$
\mathrm{x}^{2}+\mathrm{y}^{2}-2 h \mathrm{x}=0
\quad \text{...(i)}
$$
On differentiating w.r.t. $x$, we get
$$
\begin{aligned}
&2 \mathrm{x}+2 \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}-2 \mathrm{~h}=0 \\
&\Rightarrow 2 \mathrm{x}+2 \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}-\left(\frac{\mathrm{x}^{2}+\mathrm{y}^{2}}{\mathrm{x}}\right)=0 \quad \text { [from Eq. (i)] } \\
&\Rightarrow \quad \mathrm{x}^{2}-\mathrm{y}^{2}+2 \mathrm{xy} \frac{\mathrm{dy}}{\mathrm{dx}}=0
\end{aligned}
$$
Hence, the correct answer is option: 'A'.