Question

If the length of the subnormal at any point of the curve is constant, then the eccentricity of this curve is

• A. $e=\sqrt{2}$
• B. $e>1$
• C. $0<e<1$
• D. $e=1$

Answer 1

The correct option is D $e=1$

The length of subnormal is constant.
We know that, length of subnormal is given by $\left|y\frac{dy}{dx}\right|$
$\Rightarrow \left|y\frac{dy}{dx}\right|=p,p$ is a constant
$\Rightarrow y\frac{dy}{dx}=\pm p$
$\Rightarrow ydy=\pm pdx$
Integrating on both sides
$\frac{y^2}{2}=\pm px+C$
This is the equation of a parabola.
For parabola, eccentricity, $e=1$