Problem 268
Determine α,β so that the plane α x+β y+z+1=0 contains the…
Determine $\alpha,\beta$ so that the plane $\alpha x+\beta y+z+1=0$ contains the line $x-1=y-2=z-3$.
Explanation of Method
Choosing α and β so that the plane contains the given line.
Step 1: Solution
Using Problem 249, with $M_1(1,2,3)$ and $\vec d=\{1,1,1\}$: $\alpha+2\beta+4=0,\qquad \alpha+\beta+1=0$.
\[\alpha=2,\quad\beta=-3:\qquad 2x-3y+z+1=0\]
Result
\[\boxed{\,\alpha=2,\quad\beta=-3:\qquad 2x-3y+z+1=0\,}\]