Question

The equations of the curve whose subnormal is equal to a constant $a$ is

• A. $y=ax+b$
• B. $y^2=2ax+2b$
• C. $a{y}^2-x^3=a$
• D. None of these

Answer 1

The correct option is B $y^2=2ax+2b$

Therefore subnormal $NK=ycot(\psi -\frac{\pi }{2})=ycot(\frac{\pi }{2}-\psi )=-ytan\psi$

By the given condition we have $-ytan\psi =-a$

$\Rightarrow y\frac{dy}{dx}=a$

$\Rightarrow ydy=adx$

$\Rightarrow \int ydy=\int adx$

$\Rightarrow \frac{y^2}{2}=ax+b$ where $b$ is integration constant.

$\Rightarrow y^2=2ax+2b$

Thus the equations of the curve whose subnormal is equal to a constant $a$ is $y^2=2ax+2b$