Let the equation of the plane P containing the line $x + 10 = \frac{8-y}{2} = z$ be $ax + by + 3z = 2(a+b)$ and the distance of the plane P from the point $(1, 27, 7)$ be $c$. Then $a^2 + b^2 + c^2$ is equal to _____.
Step-by-Step Solution
Key Concept: The plane contains the line (so substituting point $(-10,8,0)$ and using direction $(1,-2,1)$ perpendicular to normal). Solve for $a,b$, then find distance $c$.
$a=1, b=2$. Plane: $x+2y+3z=6$. Distance from $(1,27,7)$: $c = \frac{|1+54+21-6|}{\sqrt{14}} = \frac{70}{\sqrt{14}} = 5\sqrt{14}$. $a^2+b^2+c^2 = 1+4+350 = 355$. Answer: 355
Correct Answer: 355