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

Explanation

The equation fo the given plane is `x-2y+2z-3=0`.
Let the required plane parallel to the given plane be
`x-2y+2z+lambda=0` for some real number `lambda`…………….(i)
Now, the plane (i) is at unit distance from the point (1,2,3).
`therefore (|1-(2xx2)+(2xx3)+lambda|)/(1^(2)+(-2)^(2)+2^(2)) = 1 rArr |lambda+3|=3`.
`rArr lambda+3=3` or `-(lambda+3)=3`.
`rArr lambda=0` or `lambda=-6`.
Hence, the equations of the required planes are
`x-2y+2z=0` and `x-2y+2z-6=0`.