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 ${\int }_0^4\left(2\sqrt{x}-x\right)dx=\frac{8}{3}$

$\Rightarrow$ $A_{S_2}$ $=\frac{16}{3}$ and $A_{S_1}=A_{S_3}=\frac{16}{3}$ $\{\frac{\text{Area of rectangle}-S_2}{2}\}$
$\Rightarrow$ $A_{S_1}:A_{S_2}:A_{S_2}::1:1:1$ .