I need help with the following problem:
Planes $S_1$ and $S_2$ are parallel to each other and the distance between them is $2$. Plane $S_1$ passes through the points $A=(2,0,3)$ and $B=(0,0,6)$. Plane $S_2$ passes through the point $C=(-2,0,2)$.
I Need to find the equations of both planes.
I tried to write a few equations and solve without success.
Assume the equation of $S_1$ and $S_2$ are $ax + by + cz + d_1 = 0$ and $ax + by + cz + d_2 = 0$ respectively.
Given: $2a + 3c + d_1 = 0$, $6c + d_1 = 0$ and $-2a + 2c + d_2 = 0$
Also $$\frac{|d_2 - d_1|}{\sqrt{a^2 + b^2 + c^2}} = 2$$
From the first three equations, $c = \frac{2a}{3}$, $d_1 = -4a$ and $d_2 = \frac{2a}{3}$
From the last equation, $b = \pm 2a$
Hence $S_1$ is $x \pm 2y + \frac{2}{3}z - 4 = 0$ and
$S_2$ is $x \pm 2y + \frac{2}{3}z + \frac{2}{3} = 0$