If the length of subnormal of a curve is constant and the curve passes through the origin, then find the equation of the curve.
Length of subnormal of a curve at any point $P \left(x,y\right)$ is $= y \; \dfrac{dy}{dx}$
Given: Length of subnormal $=$ constant
i.e. $y \; \dfrac{dy}{dx} = k$ $\;\;\;$ [k is a constant]
i.e. $y \; dy = k \; dx$
i.e. $\displaystyle \int y \; dy = k \int dx$
i.e. $\dfrac{y^2}{2} = k \; x + c$ $\;\;\; \cdots \; (1)$ $\;\;\;$ [c is the constant of integration]
Given: The required curve passes through the origin
$\therefore$ $\;$ Putting $x = 0$, $y = 0$ in equation $(1)$ gives
$c = 0$ $\;\;\; \cdots \; (2)$
$\therefore$ $\;$ In view of equation $(2)$, equation $(1)$ becomes
$\dfrac{y^2}{2} = k \; x$ $\implies$ $y^2 = 2 \; k \; x$ $\;\;\; \cdots \; (3)$
Equation $(3)$ is the required equation of the curve.