Focus on parabolas

In Strang I read that a parabola has a point called the focus.

Imagine that light rays are coming from the stars down a telescope to its reflective mirror, that is, vertically down to the inside surface of a parabola. All such rays are supposed to be reflected to the same point, which is the focus.

For one particular (positive) x, the vertical rays will be reflected along a horizontal line.



In the figure, I plotted a part of the parabola y = x², and also its tangent line at the point (0.5,0.25) which has a slope of 1 (2 times x).

The solution for this particular point is easy. According to the law of reflection, the angle between the vertical blue line of the incoming ray and the slope line (red) is equal to the angle between the outgoing horizontal blue line and the slope. This happens when the red line has a slope of 1 and the angle (call it θ) is π/4.

At any other point (except x = - 0.5), the angle the vertical line makes with the slope is not π/4 and the outgoing ray is not horizontal. But the claim is that the outgoing ray still passes through the point (0,0.25).

I'm going to see if I can prove that. I'll report back later.

R code:

f <- function(x) { x**2 } plot(f,from=0,to=1, col='black',lwd=5) m <- function(x) { x - 0.25} plot(m,from=0.25,to=1.5, col='red',lty=2,lwd=5,add=T) lines(c(0,0.5),c(0.25,0.25),lty=2, lwd=3,col='blue') lines(c(0.5,0.5),c(0.25,1),lty=2, lwd=3,col='blue') points(0.5,0.25,col='black',pch=16,cex=3) points(0,0.25,col='blue',pch=16,cex=3) lines(c(0,0),c(0,4),lty=2) lines(c(0,2),c(0,0),lty=2)