Question

The order of differential equation of all circles of given radius '$a$' is:

• A. $4$
• B. $2$
• C. $1$
• D. $3$

Answer 1

The correct option is B $2$

Equation of circle with centre $(h,k)$ and radius $a$ is given by $(x-h{)}^2+(y-k{)}^2=a^2\cdots \left(1\right)$
Differentiating wrt $x,$ we get
$2(x-h)+2(y-k)y^{\prime }=0$
$\Rightarrow (x-h)+(y-k)y^{\prime }=0\cdots \left(2\right)$
Differentiating $\left(2\right)$ wrt $x,$ we get
$1+(y^{\prime }{)}^2+(y-k)y^{\prime \prime }=0$
$\Rightarrow y-k=-\frac{1+(y^{\prime }{)}^2}{y^{\prime \prime }}$
Putting the value of $(y-k)$ in $\left(2\right),$ we get
$x-h=\frac{1+(y^{\prime }{)}^2}{y^{\prime \prime }}{y}^{\prime }$
Finally, substituting the value of $(x-h)$ and $(y-k)$ in $\left(1\right),$ we get
$(1+(y^{\prime }{)}^2{)}^3=(a{y}^{\prime \prime }{)}^2$
$\therefore$ The order of the differential equation is $2$.