Question

The order and degree of the differential equation of the family of circles of fixed radius $r$ with centres on the $y-axis$, are respectively

• A. 2, 2
• B. 2, 3
• C. 1, 1
• D. 3, 1
• E. 1, 2

Answer 1

Answer: (E)

The equation of family of circles of fixed radius ' $r$ ' with centres on the $y$ -axis is
$(x-0{)}^2+(y-a{)}^2=r^2\dots (i)$
On differentiating w.r.t. $x$, we get
$2x+2(y-a)\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-\frac{x}{y-a}$
$\Rightarrow (y-a)=\frac{-x}{(dy/dx)}$
On putting this value in Eq. (i), we get
$x^2+\frac{x^2}{{\left(\frac{dy}{dx}\right)}^2}=r^2$
$\Rightarrow x^2\left\{1+{\left(\frac{dy}{dx}\right)}^2\right\}$
$=r^2{\left(\frac{dy}{dx}\right)}^2$
Hence, order $\to$ highest order derivative $=1$
and degree $\to$ power of highest order derivative $=2$