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; 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 $1:1:1$

On solving, we get intersection points of $x^2=4y$ and $y^2=4x$ are $(0,0)$ and $(4,4)$



By symmetry, we observe

$S_1=S_3=\underset{0}{\overset{4}{\int }}ydx$

$=\underset{0}{\overset{4}{\int }}\frac{x^2}{4}dx={\left[\frac{x^3}{12}\right]}_0^4=\frac{16}{3}$ sq. units

Also, $S_2=16-(S_1+S_3)$

$=16-2.\frac{16}{3}=\frac{16}{3}$ sq. units

$\therefore S_1:S_2:S_3=1:1:1$