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Shortest Distance from Point to Curve

CAT 2017 Slot 1 · Quantitative Ability · Hard · Co-ordinate Geometry

This is a hard Quantitative Ability question from the CAT 2017 Slot 1 paper. It tests Co-ordinate Geometry. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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The shortest distance of the point (12,1)\left(\frac{1}{2}, 1\right) from the curve y=∣x−1∣+∣x+1∣y = |x - 1| + |x + 1| is

  1. A.

    1

  2. B.

    0

  3. C.

    2\sqrt{2}

  4. D.

    32\sqrt{\frac{3}{2}}

Answer

A

Explanation

For −1≤x≤1-1 \le x \le 1, y=−(x−1)+(x+1)=2y = -(x - 1) + (x + 1) = 2. So the curve contains the horizontal line segment y=2y = 2 for x∈[−1,1]x \in [-1, 1]. The point P(12,1)P\left(\frac{1}{2}, 1\right) has x=12x = \frac{1}{2}, which is in [−1,1][-1, 1]. The perpendicular distance from (12,1)\left(\frac{1}{2}, 1\right) to the line y=2y = 2 is ∣2−1∣=1|2 - 1| = 1.

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