CAT 2024 Slot 1 · Quant · Question 4

CAT 2024 Slot 1 previous year question on Cicles, with the answer and a step-by-step solution.

CiclesCoordinate GeometryMultiple choiceMedium
In the XY-plane, the area, in sq. units, of the region defined by the inequalities y\ge x+4 and -4\le{x}^{2}+{y}^{2}+4\left( x-y \right)\le 0 is
  1. A
    4\pi
  2. B
    2\pi
  3. C
    \pi
  4. D
    3\pi
Pick an option, then check.
Answer & solution

Answer: Option B

Complete the squares in the quadratic expression: {x}^{2}+4x+{y}^{2}-4y={\left( x+2 \right)}^{2}+{\left( y-2 \right)}^{2}-8.

The double inequality becomes -4\le{\left( x+2 \right)}^{2}+{\left( y-2 \right)}^{2}-8\le 0.

Adding 8 throughout, 4\le{\left( x+2 \right)}^{2}+{\left( y-2 \right)}^{2}\le 8.

That is the ring between two circles centred at \left( -2,2 \right), with radii 2 and 2\sqrt{2}.

Now test the line y=x+4 at the centre: putting x=-2 gives y=2, so the line passes exactly through the centre.

A line through the centre cuts the ring into two equal halves, and y\ge x+4 keeps one of them.

The full ring has area \pi{\left( 2\sqrt{2}\right)}^{2}-\pi{\left( 2 \right)}^{2}=8\pi -4\pi =4\pi.

Half of that is 2\pi square units.

Shortcut. Check whether the line passes through the centre before doing anything else. If it does, the answer is simply half the ring and no integration or arc work is needed.

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