The area of the region containing the points (x, y) satisfy-ing `4 <= x^2 + y^2 <= 2(|x| + |y|)` is

Options

A. 8 sq. units
B. 2 sq. units
C. `4pi` sq. units
D. `2pi` sq. units

Answer

A

Explanation

The points in the required region satisfy
`4lex^(2)+y^(2)le2(|x|+|y|)" …(1)"`
Since the curve (1) is symmetrical about both the axes, the required area is 4 times the area of the region in the first quadrant. Therefore, it is sufficient to sketch the region and to find the area in the first quadrant.
In the first quadrant, the curve (1) consist of two curves
`x^(2)+y^(2)ge4" "(C_(1))`
`"and "x^(2)+y^(2)-2x-2yge0" "(C_(2))`

`therefore" "` Required area =4 (area ABCDA)
=4(area of semi-cirlce ABCA)-(area of sector ADCBA)
=4(area of semi-circle ABCA)-(area of sector OADCO-area of triangle OAC)
`=4{pi-(pi-2}=8` sq. units.