Question

Obtain the differential equation of the family of circles $x^2+y^2+2gx+2fy+c=0$; where $g,f$ and $c$ are arbitrary constants.

• A. $[1+{\left(y^{{}^{\prime }}\right)}^2]y^{\prime \prime }-3y^{{}^{\prime }}{\left(y^{\prime \prime }\right)}^2=0$
• B. $[1+{\left(y^{{}^{\prime \prime }}\right)}^3]y^{}-2y^{{}^{\prime }}{\left(y^{\prime \prime }\right)}^2=0$
• C. $[1+{\left(y^{{}^{\prime \prime }}\right)}^2]y^{{\prime \prime }^{\prime }}-3y^{{}^{\prime }}{\left(y^{\prime \prime }\right)}^2=0$
• D. None of these

Answer 1

The correct option is A $[1+{\left(y^{{}^{\prime }}\right)}^2]y^{\prime \prime }-3y^{{}^{\prime }}{\left(y^{\prime \prime }\right)}^2=0$

Given:

$x^2+y^2+2gx+2fy+c=0\to \left(1\right)$

We will have to eliminate the constant g,f abd c.

So we will have to differentiate the equation thrice.

Differentiate (1) equation w.r.t ${}^{\prime }{x}^{\prime }$

$2x+2y{y}^{{}^{\prime }}+2g+2f{y}^{{}^{\prime }}=0$

Differentiate w.r.t ${}^{\prime }{x}^{\prime }$

$2+2(y{y}^{{}^{\prime \prime }}+y^{{}^{\prime }}{y}^{{}^{\prime }})+2f{y}^{{}^{\prime \prime }}=0$

$1+y{y}^{{}^{\prime \prime }}+{\left(y^{{}^{\prime }}\right)}^2+f{y}^{{}^{\prime \prime }}=0\to \left(2\right)$

Differentiate w.r.t ${}^{\prime }{x}^{\prime }$

$0+y{y}^{{}^{\prime \prime \prime }}+y^{{}^{\prime \prime }}{y}^{{}^{\prime }}+2y^{{}^{\prime }}{y}^{{}^{\prime \prime }}+f{y}^{{}^{\prime \prime \prime }}=0$

$y^{{}^{\prime \prime \prime }}(y+f)+3y^{{}^{\prime \prime }}{y}^{{}^{\prime }}=0\to \left(3\right)$

putting the value of ${}^{\prime }{f}^{\prime }$ from equation (2) in equation (3)

From equation (2) $f=\frac{-1}{y^{{}^{\prime \prime }}}(1+y{y}^{{}^{\prime \prime }}+\left({\left(y^{{}^{\prime }}\right)}^2\right))$

$y^{{}^{\prime \prime \prime }}(y-\frac{1}{y^{{}^{\prime \prime }}}(1+y{y}^{{}^{\prime \prime }}+\left({\left(y^{{}^{\prime }}\right)}^2\right)))+3y^{{}^{\prime }}{y}^{{}^{\prime \prime }}=0$

$y^{{}^{\prime \prime \prime }}\left(\frac{y{y}^{{}^{\prime \prime }}-1-y{y}^{{}^{\prime \prime }}-{\left(y^{{}^{\prime }}\right)}^2}{y^{{}^{\prime \prime }}}\right)+3y^{{}^{\prime }}{y}^{{}^{\prime \prime }}=0$

$\frac{-y^{{}^{\prime \prime \prime }}}{y^{{}^{\prime \prime }}}(1+{\left(y^{{}^{\prime }}\right)}^2)+3y^{{}^{\prime }}{y}^{{}^{\prime \prime }}=0$

$\frac{-y^{{}^{\prime \prime \prime }}(1+{\left(y^{{}^{\prime }}\right)}^2)+3y^{{}^{\prime }}{\left(y^{{}^{\prime \prime }}\right)}^2}{y^{{}^{\prime \prime }}}=0$

$-y^{{}^{\prime \prime \prime }}(1+{\left(y^{{}^{\prime }}\right)}^2)+3y^{{}^{\prime }}{\left(y^{{}^{\prime \prime }}\right)}^2=0$

$y^{{}^{\prime \prime \prime }}(1+{\left(y^{{}^{\prime }}\right)}^2)-3y^{{}^{\prime }}{\left(y^{{}^{\prime \prime }}\right)}^2=0$

Hence this is a differential equation of family of circle.

Therefore option ${}^{\prime }{A}^{\prime }$ is correct