1.

Orthogonal trajectories of the system of curves `((dy)/(dx))^(2) = (a)/(x)` areA. `9a(y+c)^(2) = 4x^(3)`B. `y + c = (-2)/(9sqrt(a))x^(3//2)`C. `y^(2)+c = (2)/(3sqrt(a)) x^(3//2)`D. `9a (y+c)^(2) = 4x^(2)`

Answer» Correct Answer - A

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


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