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CAT 2024 Slot 1 — QA Question 11

BasicsEasy

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

Answer & solution

  • 2π

  • B

    π

  • C

    4π

  • D

    3π

Solution

Hard

Rewrite the quadratic inequality by completing the square — it becomes an annulus (region between two concentric circles). Then intersect that annulus with the half-plane y≥x+4y\ge x+4; the line happens to be tangent/chord-positioned so a clean fraction of the annulus survives.

1

Complete the square in the quadratic. Group xx and yy terms.

x2+y2+4x−4y=(x+2)2+(y−2)2−8⇒ −4≤(x+2)2+(y−2)2−8≤0(substitute)⇒ 4≤(x+2)2+(y−2)2≤8\begin{aligned} &x^2+y^2+4x-4y=(x+2)^2+(y-2)^2-8\\ &\Rightarrow\ -4\le (x+2)^2+(y-2)^2-8\le 0 \quad\text{(substitute)}\\ &\Rightarrow\ 4\le (x+2)^2+(y-2)^2\le 8 \end{aligned}
2

Interpret the region. An annulus centred at (−2,2)(-2,2) with inner radius 22 and outer radius 222\sqrt2.

Centre (−2,2),  rin=2,  rout=22\begin{aligned} &\text{Centre }(-2,2),\ \ r_{\text{in}}=2,\ \ r_{\text{out}}=2\sqrt2 \end{aligned}
3

Where does the line y=x+4y=x+4 sit? Distance from centre (−2,2)(-2,2) to x−y+4=0x-y+4=0.

d=∣(−2)−(2)+4∣12+(−1)2=02=0⇒ the line passes through the centre(diameter line)\begin{aligned} &d=\dfrac{|(-2)-(2)+4|}{\sqrt{1^2+(-1)^2}}=\dfrac{0}{\sqrt2}=0\\ &\Rightarrow\ \text{the line passes through the centre} \quad\text{(diameter line)} \end{aligned}
4

Half the annulus survives. A line through the centre splits the annulus into two equal halves; y≥x+4y\ge x+4 keeps exactly one half.

Area=12(πrout2−πrin2)⇒ Area=12π(8−4)(from step 2)⇒ Area=2π\begin{aligned} &\text{Area}=\tfrac12\left(\pi r_{\text{out}}^2-\pi r_{\text{in}}^2\right)\\ &\Rightarrow\ \text{Area}=\tfrac12\pi(8-4) \quad\text{(from step 2)}\\ &\Rightarrow\ \text{Area}=2\pi \end{aligned}
Area=2π sq. units\text{Area}=2\pi\ \text{sq. units}
CAT 2024 Slot 1 QA Q11: In the XY-plane, the area, in sq. units, of the region defined by the inequalities y ≥ x + 4 and -4 ≤ x — Solution | TheCATExam