Write the equations of two mutually perpendicular planes…

Write the equations of two mutually perpendicular planes passing through the intersection of $x=y$ and $z=0$, one of which passes through $(0,4,2)$.

Explanation of Method

Two perpendicular planes through a given line, one of them through the point (0, 4, 2).

Step 1: The pencil

The line $\ell:\ x=y,\ z=0$ passes through $O$ with direction $\{1,1,0\}$. The pencil is $x-y+\lambda z=0$, with normal $\{1,-1,\lambda\}$, together with the member $z=0$.

Step 2: First plane

Through $(0,4,2)$: $-4+2\lambda=0$, so $\lambda=2$.

\[\pi_A:\ x-y+2z=0\]

Step 3: Second plane

Perpendicularity $\{1,-1,2\}\cdot\{1,-1,\mu\}=2+2\mu=0$ gives $\mu=-1$.

\[\pi_B:\ x-y-z=0\]

Check: both contain $(t,t,0)$, $\pi_A$ contains $(0,4,2)$, and $\vec n_A\cdot\vec n_B=1+1-2=0$.

$(0,4,2)$ $\ell$

$\pi_A:\ x-y+2z=0$
$\pi_B:\ x-y-z=0$

Both planes contain $\ell:\ x=y,\ z=0$ and are perpendicular to each other; $\pi_A$ passes through $(0,4,2)$.

Result

\[\boxed{\,\pi_A:\ x-y+2z=0\,}\]

\[\boxed{\,\pi_B:\ x-y-z=0\,}\]