) Find the point on the surface z = 2x − 4y + 6 that is closest to the point P0(0, 1, 0).

Question

Find the point on the surface z = 2x − 4y + 6 that is closest to the point P₀(0, 1, 0).

Answer 1

Rewrite the equation of the plane as 2x - 4y -z = -6

(0, 1, 0) is on the plane 2x - 4y - z = -4

These planes, and thus the point and plane, are |-6 - -4| /√(2² + (-4)² + (-1)²) = 2/√21 = 2√21/21 apart.

For example, if we start at (0, 1, 0) and go in the direction (2, -4, -1)/√21 by a negative of the distance, -2√21/21 (recall that the total decreases by 2) , you get

(-4/21, 1 8/21, 2/21), which, lo and behold, is on the plane 2x - 4y - z = -6.