Correct option is (A) 9a (y + c)2 = 4x3
Given differential equation of curves
\((\frac{dy}{dx})^2=\frac ax\)
⇒ \(\frac{dy}{dx}=\frac{\sqrt a}{\sqrt x}\)
Replacing \(\frac{dy}{dx}\) with \(\frac{-dy}{dx}\), we get
\(\frac{dy}{dx}=\frac{\sqrt a}{\sqrt x}\)
⇒ \(-\frac{\sqrt a}{\sqrt x}dx=dy\)
⇒ \(\frac{-1}{\sqrt a}\int\sqrt xdx=\int dy+c\)
⇒ \(\frac{-1}{\sqrt a}\times\frac23 x^{3/2}=y+c\)
⇒ \(\frac{4}{9a}x^3=(y+c)^2\) (By squaring on both sides)
⇒ 9a (y + c)2 = 4x3