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Step-by-Step Solution
Step 1: Identify the Given Line and Its Direction Ratios
The line is provided in the symmetric form
$$
\frac{x - 1}{2} \;=\; \frac{y - 2}{3} \;=\; \frac{z - 3}{4}.
$$
From this, a point on the line is $(1,\,2,\,3)$ and its direction ratios are $(2,\,3,\,4)$.
Step 2: Note the Point to be Reflected
The point to be reflected across the line is $P(2,\,3,\,5)$. Let its mirror image with respect to the given line be $R(\alpha,\,\beta,\,\gamma)$.
Step 3: Apply the Perpendicularity Condition
If $R(\alpha,\,\beta,\,\gamma)$ is the reflection of $P(2,\,3,\,5)$ on the given line, then the segment $PR$ must be perpendicular to the direction vector $(2,\,3,\,4)$ of the line. Mathematically,
$$
\bigl(\alpha - 2,\, \beta - 3,\, \gamma - 5\bigr) \cdot (2,\,3,\,4) \;=\; 0.
$$
Step 4: Expand to Find the Desired Expression
Expanding the dot product condition:
$$
2(\alpha - 2)\;+\;3(\beta - 3)\;+\;4(\gamma - 5)\;=\;0.
$$
Simplify this:
$$
2\alpha - 4 + 3\beta - 9 + 4\gamma - 20 \;=\; 0
$$
$$
2\alpha + 3\beta + 4\gamma \;=\; 4 + 9 + 20 \;=\; 33.
$$
Step 5: Conclude the Value
Hence, the value of $2\alpha + 3\beta + 4\gamma$ for the mirror image point is $\boxed{33}$.
