Question

The equation of the curve through point $\left(1,0\right)$ which satisfies the differential equation $\left(1+{\mathrm{y}}^2\right)d\mathrm{x}-\mathrm{xy}d\mathrm{y}=0$ is

• A. ${\mathrm{x}}^2+{\mathrm{y}}^2=4$
• B. ${\mathrm{x}}^2-{\mathrm{y}}^2=1$
• C. $2{\mathrm{x}}^2+{\mathrm{y}}^2=2$
• D. None of these

Answer 1

The correct option is B

${\mathrm{x}}^2-{\mathrm{y}}^2=1$

Explanation for correct option:

Given equation of the curve,

$$\begin{gathered} \Rightarrow \left(1+{\mathrm{y}}^2\right)d\mathrm{x}-\mathrm{xy}d\mathrm{y}=0 \\ \Rightarrow \frac{d\mathrm{x}}{\mathrm{x}}-\frac{\mathrm{y}}{1+{\mathrm{y}}^2}d\mathrm{y}=0 \\ \Rightarrow \log \left({\mathrm{x}}^2\right)-\frac{1}{2}\log \left(1+{\mathrm{y}}^2\right)=\mathrm{constant} \\ \Rightarrow \log \left({\mathrm{x}}^2\right)-\log \left(1+{\mathrm{y}}^2\right)=\log \left(\mathrm{C}\right) \\ \Rightarrow {\mathrm{x}}^2=\mathrm{C}\left(1+{\mathrm{y}}^2\right) \end{gathered}$$

The equation of curve passes through point $\left(1,0\right)$

Putting $\left(1,0\right)$ in the equation,

$\Rightarrow {\mathrm{x}}^2=\mathrm{C}\left(1+{\mathrm{y}}^2\right)$

$$\begin{gathered} \Rightarrow 1=C\left(1+0\right) \\ \Rightarrow C=1 \end{gathered}$$

Put the value of constant in the equation,$\Rightarrow {\mathrm{x}}^2=\mathrm{C}\left(1+{\mathrm{y}}^2\right)$

We get,

${\mathrm{x}}^2-{\mathrm{y}}^2=1$

Hence, the correct answer is option $\left(\mathrm{B}\right)$.