Question 90

A line passes through $$A(4, -6, -2)$$ and $$B(16, -2, 4)$$. The point $$P(a, b, c)$$ where $$a, b, c$$ are non-negative integers, on the line $$AB$$ lies at a distance of 21 units, from the point $$A$$. The distance between the points $$P(a, b, c)$$ and $$Q(4, -12, 3)$$ is equal to


Correct Answer: 22

Solution

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Line AB: A(4,-6,-2), B(16,-2,4). Direction: (12,4,6)=(6,2,3). |direction|=7. Unit=(6/7,2/7,3/7).

P at distance 21 from A: P=A+21·(6/7,2/7,3/7)=A+(18,6,9)=(22,0,7).

PQ distance from (22,0,7) to (4,-12,3): $$\sqrt{324+144+16}=\sqrt{484}=22$$.

The answer is $$\boxed{22}$$.

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