Question

What is the equation of the curve whose subnormal is equal to constant $a$?

• 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

$y^2=2ax+2b$

Explanation for the correct option :

Finding equation of curve .

As we know , Subnormal $=ycot\left(\phi -\frac{\mathrm{\pi }}{2}\right)=ycot\left(\frac{\mathrm{\pi }}{2}-\phi \right)=y\tan \phi$

By the given condition, we have , $y\tan \phi =a$

$$\begin{gathered} \Rightarrow y\frac{dy}{dx}=a \\ \Rightarrow ydy=adx \\ \Rightarrow \int ydy=\int adx \\ \Rightarrow \frac{y^2}{2}=ax+b(\text{where},b\text{is integration constant}) \end{gathered}$$

Thus, the equation of curve is $y^2=2ax+2b$.

Hence, the correct answer is Option(B).