The formula for finding the foot of the perpendicular from a point $(x_1,y_1)$ to the line $ax+by+c=0$ is given by:
$$\displaystyle{\frac{x − x_1}{a} = \frac{y − y_1}{b} = \frac{−(ax_1 + by_1 + c)}{a^2 + b^2}}$$
Note: I didn't include the proof for above formula as the answer would have become too long. If someone wants the proof for above formula, refer this article https://math.stackexchange.com/questions/1013230/how-to-find-coordinates-of-reflected-point.
For finding the image of the point in the same line, we just multiply the rightmost term by $2$. The image of the point is at the same distance from the line as the point itself is from the line. So, we have to multiply it by $2$.
So, the image of the point $(x_1, y_1)$ in the line $ax_1 + by_1 + c = 0$ is given by:
$$\displaystyle{\frac{x − x_1}{a} = \frac{y − y_1}{b} = \frac{−2(ax_1 + by_1 + c)}{a^2 + b^2}}$$
Applying the above formula, we get $$\displaystyle{\frac{x − 1}{1} = \frac{y − 2}{2} = \frac{−2(1 + 2*2 - 15)}{1^2 + 2^2}} = \frac{20}{5} = 4$$
$$\displaystyle{\frac{x − 1}{1} = 4} \implies x = 5$$
$$\displaystyle{\frac{y − 2}{2} = 4} \implies y = 10$$
So, answer is Option $(C)$.
