Question

The differential equation of the family of curves for which the length of the normal is equal to a constant k, is given by

• A. $y^2\frac{dy}{dx}=k^2-y^2$
  
• B. ${\left(y\frac{dy}{dx}\right)}^2=k^2-y^2$
• C. $y{\left(\frac{dy}{dx}\right)}^2=k^2+y^2$
  
• D. ${\left(y\frac{dy}{dx}\right)}^2=k^2+y^2$

Answer 1

The correct option is B ${\left(y\frac{dy}{dx}\right)}^2=k^2-y^2$

The length of normal is given by,$y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$
$\therefore y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}=k\Rightarrow y^2[1+{\left(\frac{dy}{dx}\right)}^2]=k^2\Rightarrow y^2+y^2{\left(\frac{dy}{dx}\right)}^2=k^2\Rightarrow y^2{\left(\frac{dy}{dx}\right)}^2=k^2-y^2.$