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M and N start from the same location. M travels 10 km East and then 10 km North-East. N travels 5 km South and then 4 km South-East. What is the shortest distance (in km) between M and N at the end of their travel? A. 18.00 B. 22.50 C. 20.61 D. 25.00

GATE 2016 · General Aptitude · Direction Sense · hard

Answer: The shortest distance between M and N is approximately 20.61 km. Answer: C.

  1. Find M's final coordinates: M starts at (0,0). Walks 10 km East => (10, 0). Walks 10 km NE: x += 10/sqrt(2) ~ 7.071, y += 10/sqrt(2) ~ 7.071. M final = (10 + 7.071, 7.071) = (17.071, 7.071).
  2. Find N's final coordinates: N starts at (0,0). Walks 5 km South => (0, -5). Walks 4 km SE: x += 4/sqrt(2) ~ 2.828, y -= 4/sqrt(2) ~ 2.828. N final = (2.828, -5 - 2.828) = (2.828, -7.828).
  3. Compute distance between M and N: dx = 17.071 - 2.828 = 14.243. dy = 7.071 - (-7.828) = 14.899. d = sqrt(14.243^2 + 14.899^2) = sqrt(202.86 + 221.98) = sqrt(424.84) ~ 20.61 km.