Question

In which of the following curves, the length of the normal is equal to the length of the radius vector

• A. circle
• B. parabola
• C. ellipse
• D. rectangular hyperbola

Answer 1

Answer: (A), (D)

The correct options are
A circle
D rectangular hyperbola

$\text{Length of normal}=y\sqrt{1+(\frac{dy}{dx}{)}^2}\text{Length of radius vector}=\sqrt{x^2+y^2}\Rightarrow y\sqrt{1+(\frac{dy}{dx}{)}^2}=\sqrt{x^2+y^2}\Rightarrow y^2(\frac{dy}{dx}{)}^2=x^2\Rightarrow x=\pm y(\frac{dy}{dx})\Rightarrow x=y(\frac{dy}{dx})\text{or}x=-y(\frac{dy}{dx})\Rightarrow xdx-ydy=0\text{or}xdx+ydy=0\Rightarrow x^2-y^2=C_1\text{or}{x}^2+y^2=C_2{x}^2-y^2=C_1:\text{Rectangular hyperbola}{x}^2+y^2=C_2:\text{Circle}$