JEEOlympiad
Among all planes that contain the line L: (x - 1)/2 = (y - 2)/3 = (z…
Question
Among all planes that contain the line L: (x - 1)/2 = (y - 2)/3 = (z - 3)/1, let Π be the one whose distance from the point P = (4, 0, 5) is as large as possible. This maximum distance D satisfies D^2 = m/n in lowest terms with m, n positive integers. Find m + n.
✓ Verified answer: 124checked by our engine — not a guess
Step-by-step solution
Every plane through L has distance to P at most the distance from P to the line L itself, with equality for the plane whose normal points from P to the foot M of the perpendicular onto L.
Hence D = dist(P, L).
Line point A = (1, 2, 3), direction d = (2, 3, 1). Foot M = A + t·d where d·(M - P) = 0.
A - P = (-3, 2, -2). d·(A - P) = -6 + 6 - 2 = -2; d·d = 4 + 9 + 1 = 14. So t = 2/14 = 1/7.
M = (1 + 2/7, 2 + 3/7, 3 + 1/7) = (9/7, 17/7, 22/7). M - P = (-19/7, 17/7, -13/7).
D^2 = (19^2 + 17^2 + 13^2)/49 = (361 + 289 + 169)/49 = 819/49 = 117/7.
Then m + n = 117 + 7 = 124.
Final answer124
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