Place a light source at the focus of a parabolic mirror, and the light rays are reflected into a straight parallel beam, heading away from the mirror. Conversely, light streaming in from a far away source is reflected toward the focus. This is how a large telescope concentrates and magnifies the light from a distant galaxy.
Consider the curve y = ax2, the equation of the parabola. It doesn't really matter whether we measure distance in inches or meters or miles, so scale x and y by 1/a. The new equation becomes y = x2. The focus of this parabola is on the y axis, at y = ¼.
Let p be a point on the parabola, and drop a perpendicular from p down to the directrix, and call this point q. Light leaves the focus f, strikes the parabola at p, and is reflected straight up along the line qp.
Draw a line tangent to the mirror at p, and let this line intersect the directrix at r. Now ∠fpr is the angle of incidence, and using the principle of vertical angles, ∠rpq is the angle of reflection. We need to show these angles are equal. If they are, then the light beam must be reflected along the line qp.
If the triangles qpr and rpf are congruent, then the two angles are equal. They both share the common side pr, and the segment pq equals the segment pf. (We established this in the previous section.) If the third sides are equal, the triangles are congruent, and the angle of incidence equals the angle of reflection.
Let p be the point s,s2, hence q = s,-¼. Of course the focus f is at 0,¼. We only need find r.
The tangent line through p has slope 2s. It has the following equation.
y-s2 = 2s×(x-s)
Set y = -¼, and x becomes (-¼-s2)/2s + s. Subtract this from s, and the distance from q to r is (s2+¼)/2s. For large values of s, this is about half of s, which is what we would expect.
Use the pythagorean theorem to find the distance from r to f. Rewrite x as (-¼+s2)/2s. Square this and add the square of ½, giving the following.
(¼2 + ½s2 + s4) over 4s2
Take the square root to get the distance (¼+s2)/2s, which equals the distance from q to r. The triangles are congruent, the angles are equal, and the parabola reflects light from the focus onto a distant object, or light from a distant object back to the focus. (Limerick)
Applications are not restricted to visible light. Look at any satellite dish; it has the shape of a parabola. It concentrates the signal from the transmitting satellite onto the receiver, then down to your tv.
The same principle works for sound. You've probably stood in an exploratorium whispering into a parabolic reflecter. Your sounds are reflected straight behind you, across a crowded noisy room, to another parabolic dish where your friend is standing. You can hear each other easily, despite the din.