Question

The differential equation of the family of general circles is:

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

Answer 1

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

General eqn of family of circles is
$x^2+y^2+2gx+2fy+c=0$
Differentiating w.r.t x, we get
$2x+2y{y}^{\prime }+2g+2f{y}^{\prime }=0$
$\Rightarrow (y+f)y^{\prime }=-g-x$ .....(1)
Differentiating (1) w.r.t. x, we get
$(y+f)y^{\prime \prime }+(y^{\prime }{)}^2=-1$ .....(2)
$\Rightarrow (y+f)=\frac{-1-y^{\prime 2}}{y^{\prime \prime }}$ ....(3)
Differentiating (2) w.r.t. x, we get
$\Rightarrow (y+f)y^{\prime \prime \prime }+y^{\prime \prime }{y}^{\prime }+2y^{\prime }{y}^{\prime \prime }=0$
Substituting the value of $y+f$ from (3), we get
$\Rightarrow \frac{-1-y^{\prime 2}}{y^{\prime \prime }}{y}^{\prime \prime \prime }+y^{\prime \prime }{y}^{\prime }+2y^{\prime }{y}^{\prime \prime }=0$
$\Rightarrow y^{\prime \prime \prime }(1+y^{\prime 2})+3y^{\prime }{y}^{\prime \prime 2}=0$