Parabolic reflector proof

1. Oct 28, 2008

phil ess

1. The problem statement, all variables and given/known data

Let P(x1,y1) be a point on the parabola y2 = 4px with focus F(p,0). Prove that the angles "a" and b" are equal, thus showing that a paraboloid will reflect all light from the focus in a parallel beam.

2. Relevant equations

Dont know of any

3. The attempt at a solution

unfortunately i have no idea how to start this one. can someone get me going in the right direction?

Oh heres what it looks like:

Last edited: Oct 28, 2008
2. Oct 29, 2008

Staff: Mentor

You know y1, right? From the equation of the parabola you can solve for x1.

Now, find the slope of the line tangent to the curve at (x1, y1).

The tangent of angle b is exactly the slope of the tangent line, so if you know the tangent of an angle, you can find its arctangent.

Angle b is also the angle that the tangent line makes with the negative x-axis. Look at the triangle with vertices F, P, and the point of intersection of the tangent line and x-axis. You can get the slope of segment FP. This slope is the tangent of the acute angle FP makes with the positive x-axis. Call this angle c. From angle c you can get the measure of the obtuse angle that is the supplement (adds to 180 degrees). At this point you know two interior angles of the triangle, so it's pretty easy to get the third, a.

3. Oct 29, 2008

HallsofIvy

Staff Emeritus
You will need:
[tex]tan(a+ b)= \frac{tan(a)+ tan(b)}{1+ tan(a)tan(b)}[/itex]