Question

Find the equations of the planes parallel to the plane x-2y + 2z-4 = 0, which are at a unit distance from the point (1,2, 3).

Answer 1

The equation of the planes parallel to the plane x −2y + 2z − 4 = 0
are of the form x-2y+2z+k=0

The distance of a plane ax+by+cz+ λ from a point  (x₁,y₁,z₁)  is given by

`d=|(9ax_1+by_1+cz_1+lambda)/sqrt(a^2+b^2+c^2)|`

It is given the plane x-2y+2z+k=0 at an unit distance from the point (1, 2, 3).

`d=|(1-2(2)+2(3)+k)/(sqrt(1^2+(-2)^2)+(2)^2)|`

`1=|(k+3)/3|`

∴ |k+3|=|3|
∴ k=0 or k=-6
The equation of the planes parallel to the plane x-2y+2z-4=0
are of x-2y+2z=0 and x-2y+2z=6