GATE 2021 AE – Question 6
A person travels 3 units east and then 4 units south from O to P. From P, the person travels northeast to a point 6 units east of O, then northwest to Q, which is 8 units north of P. The shortest distance between O and Q is:
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Correct answer: (C) 5
Explanation
Place the start at the origin O $=(0,0)$ with east as $+x$ and north as $+y$.
1. 3 units east, then 4 units south: P $=(3,-4)$.
2. From P the person goes north-east to the point 6 units east of O. That point is $(6,0)$: 3 units east and 4 units north of P, a leg of length 5.
3. From $(6,0)$ the person goes north-west to Q. Q is 8 units north of P, so Q $=(3,-4+8)=(3,4)$ (a leg of 3 units west and 4 north, again of length 5).
**Distance OQ:**
$$OQ=\sqrt{3^2+4^2}=\sqrt{25}=\mathbf{5}.$$
Only the position of Q matters: it is 8 units directly north of P, which puts it at $(3,4)$ and gives a 3-4-5 right triangle with O.