Interpretation of “image of a point in a line”
The image of point $P(8,5,7)$ in the given line means the reflection of $P$ about that line.

Write the line in parametric form
Given $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{5}=t$ so a general point on the line is

Find the foot of perpendicular from $P$ to the line
Let $F$ be the foot of perpendicular from $P(8,5,7)$ to the line. Then $F=(1+2t,-1+3t,2+5t).$ Since $\overrightarrow{PF}$ is perpendicular to the line direction vector, $\big(P-F\big)\cdot \vec d=0.$

Now, $P-F=(8-(1+2t),\;5-(-1+3t),\;7-(2+5t))=(7-2t,6-3t,5-5t).$

So, $(7-2t,6-3t,5-5t)\cdot(2,3,5)=0.$

$2(7-2t)+3(6-3t)+5(5-5t)=0$ $14-4t+18-9t+25-25t=0$ $57-38t=0$ $t=\frac{57}{38}=\frac{3}{2}.$

Coordinates of the foot $F$
Substitute $t=\frac32$:

Use midpoint property of reflection
If $P(8,5,7)$ is reflected in the line to $P'(\alpha,\beta,\gamma)$, then the foot $F$ is the midpoint of $PP'$. Hence $P' = 2F-P.$

Therefore, $\alpha = 2\cdot 4 - 8 = 0,$ $\beta = 2\cdot \frac72 - 5 = 7-5=2,$ $\gamma = 2\cdot \frac{19}{2} - 7 = 19-7=12.$

So, $(\alpha,\beta,\gamma)=(0,2,12).$

Compute the required sum $\alpha+\beta+\gamma=0+2+12=14.$

Check options
The correct option is: $\boxed{14}$ which is Option D.