Question

The equation of family of curves for which the length of the normal is equal to the radius vector is

• A. $y^2\pm x^2=k$
  
• B. $y\pm x=k$
  
• C. $y^2=kx$
• D. $Noneofthese$

Answer 1

The correct option is A $y^2\pm x^2=k$


Length of the normal =$y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}$
It is given that $y\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}=\sqrt{x^2+y^2}$
($\because$ Radius vector =$r=\sqrt{x^2+y^2})$
$\Rightarrow y^2+y^2{\left(\frac{dy}{dx}\right)}^2=x^2+y^2\Rightarrow y^2{\left(\frac{dy}{dx}\right)}^2=x^2\Rightarrow ydy\pm xdx=0\Rightarrow y^2\pm x^2=k.$