Question

The differential equation of the family of circles touching the $x$-axis at origin is given by

• A. $y^{\prime \prime }=\frac{1}{x^2-y^2}{y}^{\prime }$
• B. $y^{\prime }=\frac{2xy}{x^2-y^2}$
• C. $y^{\prime \prime }-y^{\prime }=\frac{xy}{x^2-y^2}$
• D. none of these

Answer 1

Answer: (B)

Family of circles represented graphically is shown below.



Equation of the family of circles is
$x^2+(y-a{)}^2=a^2$
$\Rightarrow x^2+y^2=2ay\dots \left(i\right)$
Differentiating $(i)$ w.r.t. $x$, we get
$2x+2y\frac{dy}{dx}=2a\frac{dy}{dx}$
$\Rightarrow x+y{y}^{\prime }=a{y}^{\prime }$
$\Rightarrow a=\frac{x+y{y}^{\prime }}{y^{\prime }}$
Substituting value of $‘a’$ in equation $(i)$
$x^2+y^2=2y\left(\frac{x+y{y}^{\prime }}{y^{\prime }}\right)$
$\Rightarrow x^2{y}^{\prime }+y^2{y}^{\prime }=2xy+2y^2{y}^{\prime }$
$\Rightarrow y^{\prime }=\frac{2xy}{x^2-y^2}$