How can I find the coordinates of a point reflected over a line that may not necessarily be any of the axis?
Example Question:
If P is a reflection (image) of point (3, -3) in the line $2y = x+1$, find the coordinates of Point P.
I know the answer is $(-1,5)$ by drawing a graph but other than that, I cannot provide any prior workings because I don't know how to start…
Best Answer
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: $$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{-(ax_1+by_1+c)}{a^2+b^2}$$
For finding the image of the point in the same line, we just multiply the rightmost term by 2.
So, the image of the point $(x_1,y_1)$ in the line $ax_1+by_1+c=0$ is given by: $$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{-2(ax_1+by_1+c)}{a^2+b^2}$$
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. That's what I think.
Here's the proof from my book:
Please excuse the image size.. :P