Question

The parabolas $y^{2}=4x$ and $x^{2}=4y$ divide the square region bounded by the lines $x=4$ , $y=4$ and the coordinate axes. If $S_{1},S_{2},S_{3}$ are respectively the areas of these parts numbered from top to bottom of the square region, then $S_{1}:S_{2}:S_{3}$ is

• A. $2:1:2$
• B. $1:1:1$
• C. $1:2:1$
• D. $1:2:3$

Answer 1

Answer: (B)

$y^{2}=4x$ and $x^{2}=4y$ are symmetric about line $y=x$
$\Rightarrow $ Area bounded between $y^{2}=4x$ and $y=x$ is $\displaystyle \int _{0}^{4} \left(2 \sqrt{x} - x\right) d x = \frac{8}{3}$

$\Rightarrow $ $A_{S_{2}}$ $=\frac{16}{3}$ and $A_{S_{1}}=A_{S_{3}}=\frac{16}{3}$ $\left\{\right. \frac{\text{Area of rectangle} - S_{2}}{2} \left.\right\}$
$\Rightarrow $ $A_{S_{1}}:A_{S_{2}}:A_{S_{2}}::1:1:1$ .