# What are the equations of the planes that are parallel to the plane #x+2y-2z=1# and two units away from it?

First we'll find the equation of ALL planes parallel to the original one. As a model consider this lesson:

Equation of a plane parallel to other

Or

Now we'll find planes that obey the previous formula and at a distance of 2 units from a point in the original plane. (We should expect 2 results, one for each half-space delimited by the original plane.) As a model consider this lesson:

Distance between 2 parallel planes

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The equation of a plane parallel to the plane ( x + 2y - 2z = 1 ) and two units away from it is given by:

[ x + 2y - 2z = d ]

where ( d ) is the distance from the original plane. Since the new plane is two units away, ( d = 1 + 2 = 3 ). Thus, the equation of the plane is:

[ x + 2y - 2z = 3 ]

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