Question
Quantitative Aptitude Question on Coordinate Geometry
In the XY-plane, the area, in sq. units, of the region defined by the inequalities y≥x+4 and −4≤x2+y2+4(x−y)≤0 is
A
2π
B
3π
C
π
D
4π
Answer
2π
Explanation
Solution
Consider the second inequality:
−4≤x2+y2+4(x−y)≤0.
We can rewrite the second inequality:
x2+y2+4x−4y≤4,
x2+y2+4x−4y+4≤8,
(x+2)2+(y−2)2≤8.
This represents a circle centered at (−2,2) with radius 8=22.
Now, combine the first inequality y≥x+4, which represents the region above the line y=x+4.
The area of the region is the area of the circle segment cut off by the line.
This can be calculated as half of the circle, since the line y=x+4 divides the circle into two equal parts.
The area of the circle is π×(22)2=8π. Therefore, the area of the region is:
28π=4π.
The area defined by the inequalities is 2π.