The plane x/1 + y/2 + z/3 = 1 intersects the x-axis, y-axis, and z-axis at points A, B, and C respectively. If k denotes the distance from the origin to the orthocenter of triangle ABC, find the value of 7k.

Correct answer: 6

Solution

A=(1,0,0), B=(0,2,0), C=(0,0,3). Solving the orthocenter conditions gives H=(36/49,18/49,12/49), with |OH|=42/49=6/7, so 7k = 6.

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