TITLE: Determining a coefficient in equations for planes, for which the planes do not have any point in common
QUESTION [0 upvotes]: I have four planes in an ordinary orthogonal
(x, y, z)-coordinate system – given by the equations:
$$
\begin{cases}
α_1 :& x &+ 2ay + az &= a^2 \\
α_2 :& x &+ ay + az &= a  \\
α_3 :& x &+ a^2 z &= a^3 \\
α_4 :& ax &+ ay + a^2 z &= a \\
\end{cases}
$$
I'm asked to find a value for (a), for which the four planes do not have any point in common. What steps do you recommend taking towards a solution?

REPLY [0 votes]: Consider following system os equations:
$$\begin {cases} Ax+By+Cz=D \\A_1x+B_1y+C_1z=D_1\end{cases}$$
If:
$$\frac{A_1}A=\frac{B_1}B=\frac{C_1}C=\frac{D_1}D$$
then the system is consistent and it represent a line. Now we find thses for for equations:
1&2: $1/1=1,2a/a=2, 1/1=1, a^2/a=a$ or $(1, 2, 1, a)$
2&3: $(1, 0, a, a^2)$
3&4: $(1/a, 0, a, a^2)$
1&4: $(a, 1/2, a, 1/a)$
1&3: $(1, 0, a, a)$
2&4: $(a, 1, a, 1)$
As can be seen all systems of couple of plains are not consistent for any $a$ except the system of the couple 2&4 which can be consistent if $a=1$. Hence the condition is $a\neq 1$
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