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Question:

An equation of a plane parallel to the plane x - y + 2z = 0 and at a unit distance from the origin is:

x - y + 2z - 1 = 0

x - y + 2z - 3 = 0

x - y + 2z + 5 = 0

x - y + 2z + 1 = 0

Solution:

Let an equation of a plane parallel to the plane x - y + 2z = 0 be x - y + 2z + k = 0. (i)
Perpendicular distance from O(0,0,0) to (1) is
1 = |k| / √(1 + 4 + 4) = 1
=> |k| = 3
=> k = ±3
Therefore, x - y + 2z ± 3 = 0
Hence, x - y + 2z - 3 = 0 or x - y + 2z + 3 = 0