Question

The ratio in which the area bounded by the curves $y^2=12x$ and $x^2=12y$ is divided by the line $x=3$ is

• A. $15:49$
• B. $13:37$
• C. $15:23$
• D. $17:50$

Answer 1

Answer: (A)

We have $y^2=12x$ and $x^2=12y$, parabolas with vertex $\left(0,0\right)$.

The curves intersect at $x=0,12$
Required ratio $=\frac{{\int }_0^3\left(\sqrt{12x}-\frac{x^2}{12}\right)dx}{{\int }_3^{12}\left(\sqrt{12x}-\frac{x^2}{12}\right)dx}=\frac{{\left|\sqrt{12}\cdot \frac{2}{3}{x}^{\frac{3}{2}}-\frac{x^3}{36}\right|}_0^3}{{\left|\sqrt{12}\cdot \frac{2}{3}{x}^{\frac{3}{2}}-\frac{x^3}{36}\right|}_3^{12}}$
$=\frac{15}{49}$ sq. units