Question

The differential equation of the family of general circles is

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

Answer 1

Answer: (A)

The equation of the general circle is given by
$x^{2}+y^{2}+2 g y+2 f y+c=0...$(1)
Differentiating with respect to $x$, we get
$2 x+2 y y^{\prime}+2 g+2 f y^{\prime}=0...$(2)
Differentiating again, we get
$1+y^{\prime 2}+y y^{\prime \prime}+f y^{\prime \prime}=0...$(3)
Differentiating again, we have
$2 y^{\prime} y^{\prime \prime}+y y^{\prime \prime \prime}+y^{\prime} y^{\prime \prime}+f y^{\prime \prime \prime}=0...$(4)
Eliminating $f$ from (3) and (4), we get
$y^{\prime \prime \prime}\left(1+y y^{\prime \prime}+y^{\prime 2}\right)-y^{\prime \prime}\left(y y^{\prime \prime \prime}+3 y^{\prime} y^{\prime \prime}\right)=0$
$\Rightarrow y^{\prime \prime \prime}\left(1+y^{\prime 2}\right)-3 y^{\prime} y^{\prime \prime 2}=0$,
which is the required differential equation.