Now, we can proceed with our actual proof.
To prove: A parabola will focus all rays coming from infinite, and parallel to its axis, to a single point which is its focus.
In the image to the left, the red line is the parabola, the point F is the focus and the line L is the directrix. Thus, every point on the red line is equidistant from F and L. The axis of this parabola is the vertical line.
Now, we need only prove that all rays coming parallel to the axis need to travel the same distance to reach F.
Let us consider another line M parallel to L. All rays parallel to the axis originating at the same time will reach line M at the same time. Thus, they will also reach L at the same time.
Now, notice that the length FP = PQby the definition of parabola.
For any arbitrary ray,
Distance from M to L = RP + PQ = RP + FP.
So, if rays reach L at the same time, they must reach F also at the same time as the distance is same. Thus, all rays coming from infinite and parallel to the axis will focus at F.