Question

The differential equation of the family of circles with fixed radius $r$ and with centre on $y-axis$ is

• A. $y^2(1+y_1^2)=r^2{y}_1^2$
• B. $y^2=r^2{y}_1+y_1^2$
• C. $x^2(1+y_1^2)=r^2{y}_1^2$
• D. $x^2=r^2{y}_l+y_1^2$

Answer 1

Answer: (C)

The equation of the family of circles with fixed radius $r$ and centre on $y$-axis is
$x^2+{\left(y-K\right)}^2=r^2\dots \left(1\right)$
where $K$ is a parameter.
Diff. $\left(1\right)$ both sides w.r.t. $‘x’$, we get
$2x+2\left(y-K\right)\frac{dy}{dx}=0$
i.e., $x+\left(y-K\right)\frac{dy}{dx}=0\dots \left(2\right)$
$\therefore y-K=-\frac{x}{y_1}\dots \left(3\right)$
Putting in $\left(1\right)$, we get $\left[\text{Where}{y}_1=\frac{dy}{dx}\right]$
$x^2+\frac{x^2}{y_1^2}=r^2$
$\Rightarrow x^2\left(1+y_1^2\right)=r^2{y}_1^2$