Find the line p parallel to the plane x+y=0 that meets the…

Find the line $p$ parallel to the plane $x+y=0$ that meets the lines below at two points $3$ apart: $p_1:\ \frac x3=\frac{y+1}1=\frac{z-3}{-2},\qquad p_2:\ y=z+2,\ x=1$.

Explanation of Method

A line parallel to a plane that cuts two given lines at points exactly 3 apart.

Step 1: Setup

\[ A=(3s,\,-1+s,\,3-2s)\in p_1,\qquad B=(1,\,2+t,\,t)\in p_2, \]

\[ \overrightarrow{AB}=\{1-3s,\ 3+t-s,\ t-3+2s\}. \]

Step 2: Parallel to $x+y=0$

\[ (1-3s)+(3+t-s)=0\ \Longrightarrow\ t=4s-4,\qquad \overrightarrow{AB}=\{1-3s,\ 3s-1,\ 6s-7\}. \]

Step 3: Length 3

\[ 2(1-3s)^2+(6s-7)^2=9\ \Longrightarrow\ 54s^2-96s+42=0\ \Longrightarrow\ 9s^2-16s+7=0, \]

\[ s_1=1,\qquad s_2=\tfrac79 . \]

Step 4: Two solutions

$s=1,\ t=0$: $A(3,0,1)$, $B(1,2,0)$, $\overrightarrow{AB}=\{-2,2,-1\}$, of length $3$.

\[\frac{x-3}{-2}=\frac{y}{2}=\frac{z-1}{-1}\]

$s=\tfrac79,\ t=-\tfrac89$: $A\left(\tfrac73,-\tfrac29,\tfrac{13}9\right)$, $B\left(1,\tfrac{10}9,-\tfrac89\right)$, $\overrightarrow{AB}=\left\{-\tfrac43,\tfrac43,-\tfrac73\right\}=\tfrac13\{-4,4,-7\}$, of length $\tfrac13\sqrt{81}=3$.

\[\frac{x-1}{-4}=\frac{y-\frac{10}9}{4}=\frac{z+\frac89}{-7}\]

Checks: $-4+4=0$, so the line is parallel to the plane. $B$ satisfies $y=z+2$ and $x=1$. $\checkmark$

$A_1$ $B_1$ $A_2$ $B_2$ $p_1$ $p_2$

$s=1:\ \{-2,2,-1\}$
$s=\tfrac79:\ \{-4,4,-7\}$

The two solutions: both segments have length $3$ and are parallel to the plane $x+y=0$, but their directions differ.

Result

\[\boxed{\,\frac{x-3}{-2}=\frac{y}{2}=\frac{z-1}{-1}\,}\]

\[\boxed{\,\frac{x-1}{-4}=\frac{y-\frac{10}9}{4}=\frac{z+\frac89}{-7}\,}\]

Remark

Correction: the source first used a wrong $y$-component ($2s-1$ instead of $3s-1$); it caught this later. Its second solution is still wrong: $\overrightarrow{AB}$ is $\tfrac13\{-4,4,-7\}$, not $\{-\tfrac49,\tfrac49,-\tfrac29\}$. The claim that "both lines have direction $\{-2,2,-1\}$" is false: its own vector $\{-\tfrac49,\tfrac49,-\tfrac29\}$ has length $\tfrac23$, not $3$. The correct second line has direction $\{-4,4,-7\}$.