Question

The differential equation of the family of circles having centre on $Y$-axis and radius 3 units is

• A. $\left(x^2-9\right){\left(y^{\prime }\right)}^2+x^2=0$
• B. $\left(x^2+9\right){\left(y^{\prime }\right)}^2+x^2=0$
• C. $\left(x^2-9\right)y^{\prime }+y^2=0$
• D. $\left(x^2+9\right)y^{\prime }+y^2=0$

Answer 1

Answer: (A)

The equation of the family of circle having centre on $y$-axis and radius 3 units is
$x^2+(y-b{)}^2=\mathrm{9.....}$(i)
On differentiating Eq.(i) w.r.t. $x$, we get

$2x+2(y-b)y^{\prime }=0$
$\Rightarrow y-b=-\frac{x}{y^{\prime }}....$(ii)
On substituting this value of $(y-b)$ in Eq. (i), we get
$x^2+{\left(-\frac{x}{y^{\prime }}\right)}^2=9$
$\Rightarrow x^2\left[{\left(y^{\prime }\right)}^2+1\right]=9{\left(y^{\prime }\right)}^2$
$\Rightarrow \left(x^2-9\right){\left(y^{\prime }\right)}^2+x^2=0$
which is the required differential equation.