Let ( , , ) be the mirror image of the point (2,3,5) in the line (x-1)/(2)=(y-2)/(3)=(z-3)…

Question

Let $(\alpha ,\beta ,\gamma )$ be the mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Then, $2\alpha +3\beta +4\gamma $ is equal to

Accepted Answer

General point $(2\lambda+1,\ 3\lambda+2,\ 4\lambda+3)$. Vector from $(2,3,5)$: $(2\lambda-1,\ 3\lambda-1,\ 4\lambda-2)$, perpendicular to $(2,3,4)$:
$$4\lambda-2+9\lambda-3+16\lambda-8=0\implies 29\lambda=13\implies\lambda=\frac{13}{29}.$$
$$M=\left(\frac{55}{29},\frac{97}{29},\frac{139}{29}\right).$$
$$(\alpha,\beta,\gamma)=2M-(2,3,5)=\left(\frac{52}{29},\frac{107}{29},\frac{133}{29}\right).$$
$$2\alpha+3\beta+4\gamma=\frac{104+321+532}{29}=\frac{957}{29}=33.$$

Solutions · 1