Question

The differential equation of the family of general circles is

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

Answer 1

Answer: (A)

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