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CAT 2017 Slot 1 — QA Question 30

Domain & RangeEasy

The shortest distance of the point (1/2,1) from the curve y = |x - 1| + |x + 1| is

Answer & solution

Correct answer: 1

  • 1

  • B

    0

  • C

    √2

  • D

    √32

Solution

Easy

Resolve the curve y=∣x−1∣+∣x+1∣y=|x-1|+|x+1| piecewise. The point (12,1)\left(\tfrac12,1\right) has xx in [−1,1][-1,1], where the curve is the flat segment y=2y=2. The nearest point is straight up, so the distance is just the vertical gap.

(½, 1) d = 1
1

Describe the curve piecewise. Splitting at x=−1x=-1 and x=1x=1:

y=−2xfor x≤−1y=2for −1≤x≤1y=2xfor x≥1\begin{aligned} &y = -2x &&\text{for } x \le -1\\ &y = 2 &&\text{for } -1 \le x \le 1\\ &y = 2x &&\text{for } x \ge 1 \end{aligned}
2

Locate the point relative to the curve. Since x=12x=\tfrac12 lies in [−1,1][-1,1], the closest part of the curve is the horizontal segment y=2y=2. The nearest point on it is directly above (12,1)\left(\tfrac12,1\right), namely (12,2)\left(\tfrac12,2\right).

nearest point=(12, 2)(foot of perpendicular to y=2)\begin{aligned} &\text{nearest point} = \left(\tfrac12,\,2\right) \quad\text{(foot of perpendicular to }y=2\text{)} \end{aligned}
3

Compute the distance. Purely vertical.

d=∣2−1∣=1\begin{aligned} &d = |2 - 1| = 1 \end{aligned}
d=1d = 1

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CAT 2017 Slot 1 QA Q30: The shortest distance of the point (1/2,1) from the curve y = |x - 1| + |x + 1| is — Solution | TheCATExam