The equation of the planes parallel to the plane

Question:

The equation of the planes parallel to the plane $x-2 y+2 z-3=0$ which are at unit distance from the point $(1,2,3)$ is $a x+b y+c z+d=0$. If $(b-d)=K(c-a)$, then the positive value of $K$ is

Solution:

Let plane is $x-2 y+2 z+\lambda=0$

distance from $(1,2,3)=1$

$\Rightarrow \frac{|\lambda+3|}{5}=1 \Rightarrow \lambda=0,-6$

$\Rightarrow a=1, b=-2, c=2, d=-6$ or 0

$\mathrm{b}-\mathrm{d}=4$ or $-2, \mathrm{c}-\mathrm{a}=1$

$\Rightarrow \mathrm{k}=4$ or $-2$