CAT 2024 Slot 1 QA Q4 Coordinate Geometry

Question 4 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

In the \(X Y\)-plane, the area, in sq. units, of the region defined by the inequalities \(y \geq x+4\) and \(-4 \leq x^2+y^2+4(x-y) \leq 0\) is

A
\(4 \pi\)
B
\(2 \pi\)
C
\(\pi\)
D
\(3 \pi\)

Answer: (B) 2 π Coordinate Geometry MCQ

Solution

Complete the squares: \(x^2 + y^2 + 4x - 4y = (x+2)^2 + (y-2)^2 - 8\).
So \(-4 \le (x+2)^2 + (y-2)^2 - 8 \le 0\) means \(4 \le (x+2)^2 + (y-2)^2 \le 8\): a ring centred at \((-2, 2)\) with radii \(2\) and \(2\sqrt2\).
The line \(y = x + 4\) passes through the centre \((-2, 2)\), so \(y \ge x + 4\) keeps exactly half of the ring.
Area \(= \frac12\left(\pi \cdot 8 - \pi \cdot 4\right) = 2\pi\).

Shortcut. Always check whether the line passes through the circle's centre: if it does, the answer is simply half the area.