Question

The area of the region bounded by $y^2=x$ and x=36 is divided in the ratio 1:7 by the line x=a. then a equals-

• A. 7
• B. 8
• C. 9
• D. 0

Answer 1

The correct option is C 9

From the attached fig.,

Since the line $x=a$ divides the region $CBOAD$ in the ratio $1:7$.

Therefore,

$ar\left(ABCD\right)=7ar\left(OAB\right)$

$2\times {\int }_a^{36}ydx=2\times \left(7{\int }_0^{a}ydx\right)$

${\int }_a^{36}\sqrt{x}dx=7{\int }_0^a\sqrt{x}dx(\because y^2=x)$

${\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_a^{36}=7{\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]}_0^a$

$\Rightarrow {\left(36\right)}^{\frac{3}{2}}-a^{\frac{3}{2}}=7\times a^{\frac{3}{2}}$

$\Rightarrow 216-a^{\frac{3}{2}}=7a^{\frac{3}{2}}$

$\Rightarrow 216=8a^{\frac{3}{2}}$

$\Rightarrow a^{\frac{3}{2}}=\frac{216}{8}$

$\Rightarrow a={\left(27\right)}^{\frac{2}{3}}=9$

Hence the value of $a$ is $9$.