Question

If area bounded by the curves $y^2=4ax$ and $y=mx$ is $\frac{a^3}{3}$, then the value of m is

• A. 2
• B. -2
• C. $\frac{1}{2}$
• D. None of these

Answer 1

The correct option is A 2

The two curves $y^2=4ax$ and y=mx intersect at $(\frac{4a}{m^2},\frac{4a}{m})$ and the area enclosed by the two curves is given by ${\int }_0^{\frac{4a}{m^2}}(\sqrt{4ax}-mx)dx.$
$\therefore {\int }_0^{\frac{4a}{m^2}}(\sqrt{4ax}-mx)dx=\frac{a^2}{3}$
$\Rightarrow \frac{8}{3}\frac{a^2}{m^3}=\frac{a^2}{3}\Rightarrow m^3=8\Rightarrow m=2$.