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)

$1,2$

The equation of family of circles of fixed radius ${}^{\prime }{r}^{\prime }$ with centres on the $y$-axis is
$(x-0{)}^2+(y-a{)}^2=r^2$ .... (i)
On differentiating w.r.t. $x$, we get
$2x+2(y-a)\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-\frac{x}{y-a}$
$\Rightarrow (y-a)=\frac{-x}{\left(dy/dx\right)}$
On putting this value in Eq. (i), we get
$x^2+\frac{x^2}{{\left(\frac{dy}{dx}\right)}^2}=r^2$
$\Rightarrow x^2\{1+{\left(\frac{dy}{dx}\right)}^2\}=r^2{\left(\frac{dy}{dx}\right)}^2$
Hence, order $\to$ highest order derivative $=1$
and degree $\to$ power of highest order derivative $=2$.