Question

The differential equation of the system of circles touching the y-axis at origin is

• A. $\frac{d\mathrm{y}}{d\mathrm{x}}=\frac{\left({\mathrm{x}}^2+{\mathrm{y}}^2\right)}{2\mathrm{xy}}$
• B. $\frac{d\mathrm{y}}{d\mathrm{x}}=\frac{\left({\mathrm{y}}^2-{\mathrm{x}}^2\right)}{\mathrm{xy}}$
• C. $\frac{d\mathrm{y}}{d\mathrm{x}}=\frac{\left({\mathrm{y}}^2-{\mathrm{x}}^2\right)}{2\mathrm{xy}}$
• D. $\frac{d\mathrm{y}}{d\mathrm{x}}=\frac{\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)}{2\mathrm{xy}}$

Answer 1

The correct option is C

$\frac{d\mathrm{y}}{d\mathrm{x}}=\frac{\left({\mathrm{y}}^2-{\mathrm{x}}^2\right)}{2\mathrm{xy}}$

Explanation for the correct option:

Given: The circle touches$\mathrm{y}$-axis at origin.
The equation of circle that touches$\mathrm{y}$-axis at origin and has radius $a$ is,
$$\begin{gathered} (x-a{)}^2+y^2=a^2 \\ \Rightarrow x^2+y^2-2ax=0 \end{gathered}$$
On differentiating with respect to $\mathrm{x}$
$$\begin{gathered} 2x+2y\left(\frac{dy}{dx}\right)-2a=0 \\ a=x+y\left(\frac{dy}{dx}\right) \end{gathered}$$
On substituting $a$ in the equation,
$$\begin{gathered} x^2+y^2-2x\left(x+y\left(\frac{dy}{dx}\right)\right)=0 \\ \Rightarrow x^2+y^2-2x^2-2xy\left(\frac{dy}{dx}\right)=0 \\ \Rightarrow y^2-x^2=2xy\left(\frac{dy}{dx}\right) \\ \Rightarrow \frac{dy}{dx}=\frac{y^2-x^2}{2xy} \end{gathered}$$
Hence, the correct answer is option(C)