1. Interpret the problem

We need the mirror image of the point $P=(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}.$

Let the image point be $P'=(\alpha,\beta,\gamma).$

For reflection in a line in 3D, the given line is the perpendicular bisector line of the segment joining $P$ and $P'$. Equivalently, if $M$ is the foot of the perpendicular from $P$ to the line, then $M$ is also the midpoint of $PP'$.

So we will:

• find the foot $M$ of perpendicular from $P$ to the line,
• then use midpoint relation to get $P'$.

2. Write the line in parametric form

Given $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}=t,$ so a general point on the line is $Q(t)=(1+2t,\;2+3t,\;3+4t).$

Direction vector of the line is $\vec d=(2,3,4).$

3. Find the foot of perpendicular from $P$ to the line

Let the foot be $M=(1+2t,\;2+3t,\;3+4t).$

Since $PM$ is perpendicular to the line direction $\vec d$, we must have $\overrightarrow{PM}\cdot \vec d=0.$

Now, $\overrightarrow{PM}=M-P=((1+2t)-2,\;(2+3t)-3,\;(3+4t)-5)$ $=(-1+2t,\;-1+3t,\;-2+4t).$

Dot with $(2,3,4)$: $(-1+2t)2+(-1+3t)3+(-2+4t)4=0.$

Compute: $-2+4t-3+9t-8+16t=0$ $29t-13=0$ $t=\frac{13}{29}.$

Hence, $M=\left(1+2\cdot\frac{13}{29},\;2+3\cdot\frac{13}{29},\;3+4\cdot\frac{13}{29}\right)$ $=\left(\frac{55}{29},\;\frac{97}{29},\;\frac{139}{29}\right).$

4. Use midpoint relation to find the image point

Since $M$ is the midpoint of $P=(2,3,5)$ and $P'=(\alpha,\beta,\gamma)$, $\alpha=2M_x-2,\quad \beta=2M_y-3,\quad \gamma=2M_z-5.$

So, $\alpha=2\cdot\frac{55}{29}-2=\frac{110}{29}-\frac{58}{29}=\frac{52}{29},$ $\beta=2\cdot\frac{97}{29}-3=\frac{194}{29}-\frac{87}{29}=\frac{107}{29},$ $\gamma=2\cdot\frac{139}{29}-5=\frac{278}{29}-\frac{145}{29}=\frac{133}{29}.$

Thus, $P'=\left(\frac{52}{29},\frac{107}{29},\frac{133}{29}\right).$

5. Compute $2\alpha+3\beta+4\gamma$

$2\alpha+3\beta+4\gamma=2\cdot\frac{52}{29}+3\cdot\frac{107}{29}+4\cdot\frac{133}{29}.$

$=\frac{104+321+532}{29}$ $=\frac{957}{29}=33.$

6. Check the options

The value is $33.$ So the correct option is:

C: 33

7. Compare with stored correct answer

Stored correct answer = C.

Our derived answer is also C. Hence, they agree.