Question

What are the distance of the planes parallel to xz plane and at a distance a from it?

Answer 1

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

The normal vector is:

→n=<1,2−2>

The equation of the plane parallel to the original one passing through P(x0,y0,z0) is:

→n⋅< x−x0,y−y0,z−z0>=0

<1,2,−2>⋅<x−x0,y−y0,z−z0>=0

x−x0+2y−2y0−2z+2z0=0

x+2y−2z−x0−2y0+2z0=0

Or

x+2y−2z+d=0 [1]

where a=1, b=2, c=−2 and d=−x0−2y0+2z0

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: