Let P(x1, y1) be a point on the parabola, and PF be the line from P to the focus. Construct the line L through P parallel to the x-axis and the line M tangent to the parabola at P. The angle between L and M is b, and the angle between PF and M is

a. The angle a is the angle at which a ray from F strikes the parabola at P. Because the angle of incidence equals the angle of reflection, the ray reflected from P must be at an angle a to M. Thus, if we can show that a =

b, we have demonstrated that rays reflected from the parabola starting at F will be parallel to the x-axis.

a. First show that tan b = (p/y1). Hint: Recall from trigonometry that the slope of a line is equal to the tangent of the angle the line makes with the positive x-direction.

Also recall that the slope of the line tangent to a curve at a given point is equal to the derivative of the curve at that point.

b. Now show that tan a = (p/y1), which demonstrates that a =

b. Hint: Recall from trigonometry that the formula for the tangent of the difference between two angles a1 and a2 is tan(a2 - a1) = (tan a2 - tan a1)/(1 + tan a2 * tan a1).