Short webpage title: Proving Equal Angles on a Parabolic Reflector

[phil ess]

Homework Statement

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.

Homework Equations

Dont know of any

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 here's what it looks like:

 

[Mark44]

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.

 

[HallsofIvy]

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