Question

Find the equations of the planes that are parallel to the plane x + y + z = 0 and 4 √ 3 units away from it. (Hint: What does it mean for planes to be parallel?)

Answer 1

x+y+z+12 =0 or

x+y+z-12=0

Explanation:

Two planes are parallel if the normals are parallel.

In other words if normal of the plane has the same direction ratios or proportional direction ratios the planes are parallel.

Here one plane has equation as x+y+z =0

Normal has DR (1,1,1)

So the second plane would have equation of the form

x+y+z+k=0 such that distance between them is
`4√(3)`

Distance between these two parallel planes in terms of k is given by

=
`|(k-0)/(√(1^2+1^2+1^2) )|\\ =(k)/(√(3) ) =4√(3) \\|k| = 12`

k =12 or -12

There will be two planes as

x+y+z+12 =0 or

x+y+z-12=0