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A point Q moves along the line r = (1, 2, 0) + t(1, 1, 1). At the…
Question
A point Q moves along the line r = (1, 2, 0) + t(1, 1, 1). At the unique position where Q is equidistant from A = (3, 0, 1) and B = (1, 4, 5), find the sum of the coordinates of Q.
✓ Verified answer: 8checked by our engine — not a guess
Step-by-step solution
Q = (1 + t, 2 + t, t). The condition |Q - A|^2 = |Q - B|^2 is linear in t.
|Q - A|^2 = (t - 2)^2 + (t + 2)^2 + (t - 1)^2.
|Q - B|^2 = (t)^2 + (t - 2)^2 + (t - 5)^2.
Set equal and expand: (t-2)^2 cancels. Left rest: (t+2)^2 + (t-1)^2 = 2t^2 + 2t + 5. Right rest: t^2 + (t-5)^2 = 2t^2 - 10t + 25.
Thus 2t + 5 = -10t + 25 → 12t = 20 → t = 5/3.
Q = (1 + 5/3, 2 + 5/3, 5/3) = (8/3, 11/3, 5/3). Sum = (8 + 11 + 5)/3 = 24/3 = 8.
Final answer8
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