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(Example: $S_1$ is the area bounded by $y=4$ and $x^2=4y$ ); then $S_1,S_2,S_3$ is

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

Answer 1

The correct option is D $1:1:1$

Since $y^2=4x$ and $x^2=4y$
are symmetric about the line y = x
Now, Area bounded between $y^2=4xandy=x$ is
${\int }_0^4(2\sqrt{x}-x)dx$
$=\frac{8}{3}$
$A_{s_2}=\frac{16}{3}$ and
$A_{s_1}=A_{s_3}=\frac{16}{3}$
$A_{s_1}:A_{s_2}:A_{s_3}::1:1:1$