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CAT 2023 Slot 2 — QA Question 19

LinesEasy

The area of the quadrilateral bounded by the Y-axis, the line x = 5 and the lines |x - y| - |5 - x| = 2, is

Answer & solution

Answer: 45

Solution

Easy

Between the YY-axis and x=5x=5 we have 0≤x≤50\le x\le 5, so 5−x≥05-x\ge 0 and ∣5−x∣=5−x|5-x|=5-x. That collapses the modulus equation into two straight lines. Find where they meet x=0x=0 and x=5x=5, and the four corners form a trapezium.

1

Simplify the modulus on the strip 0≤x≤50\le x\le 5. Here ∣5−x∣=5−x|5-x|=5-x, so:

∣x−y∣−(5−x)=2⇒ ∣x−y∣=7−x⇒ x−y=±(7−x)\begin{aligned} &|x-y|-(5-x)=2\\ &\Rightarrow\ |x-y|=7-x\\ &\Rightarrow\ x-y=\pm(7-x) \end{aligned}
2

Split into the two lines:

x−y=+(7−x) ⇒ y=2x−7x−y=−(7−x) ⇒ y=7\begin{aligned} &x-y=+(7-x)\ \Rightarrow\ y=2x-7\\ &x-y=-(7-x)\ \Rightarrow\ y=7 \end{aligned}
3

Read off the four corners on x=0x=0 and x=5x=5:

x=0: y=2(0)−7=−7  and  y=7x=5: y=2(5)−7=3  and  y=7⇒ (0,−7), (0,7), (5,7), (5,3)\begin{aligned} &x=0:\ y=2(0)-7=-7 \ \text{ and }\ y=7\\ &x=5:\ y=2(5)-7=3 \ \text{ and }\ y=7\\ &\Rightarrow\ (0,-7),\,(0,7),\,(5,7),\,(5,3) \end{aligned}
(0,-7) (0,7) (5,7) (5,3) x=5
4

Area of the trapezium. The two vertical sides are parallel: left side from −7-7 to 77 has length 1414, right side from 33 to 77 has length 44, and the gap between them is 55:

Area=12(14+4)×5=12×18×5=45\begin{aligned} &\text{Area}=\tfrac12(14+4)\times 5\\ &=\tfrac12\times 18\times 5=45 \end{aligned}
Area=45\text{Area}=\mathbf{45}
CAT 2023 Slot 2 QA Q19: The area of the quadrilateral bounded by the Y-axis, the line x = 5 and the lines |x - y| - |5 - x| = 2, is — Solution | TheCATExam