Answer:
.
Step-by-step explanation:
Consider the line that is perpendicular to and goes through .
Both and the reflection would be on this new line. Besides, the two points would be equidistant from the intersection of this new line and line .
Hence, if the vector between and that intersection could be found, adding twice that vector to would yield the coordinates of the reflection.
Since this new line is perpendicular to line , the slope of this new line would be .
Hence, would be a direction vector of this new line.
(a constant multiple of would also be a direction vector of this new line.)
Both and the aforementioned intersection are on this new line. Hence, their position vectors would differ only by a constant multiple of a direction vector of this new line.
In other words, for some constant , would be the position vector of the reflection of (the position vector of is .)
would be the coordinates of the intersection between the new line and . would be the vector between and that intersection.
Since that intersection is on the line , its coordinates should satisfy:
.
Solve for :
.
.
Hence, the vector between the position of and that of the intersection would be:
.
Add twice the amount of this vector to position of to find the position of the reflection, .
-coordinate of the reflection:
.
-coordinate of the reflection:
.