Quadratics - the Australian Mathematical Sciences Institute

{32} • Quadratics
reflected ray
normal
light ray
0
Note that this implies that the angle between the ray and the tangent is also preserved
after reflection, which is a more convenient idea for us here.
Let P(2ap, ap 2 ) be a point on the parabola x 2 = 4ay with focus at S and let T be the point
where the tangent at P cuts the y-axis.
Suppose PQ is a ray parallel to the y-axis. Our aim is to show that the line PS will satisfy
the reflection property, that is, ∠QPB is equal to ∠SPT .
x2 = 4ay
y
Q
B
P(2ap‚ap2)
S(0,a)
x
tangent at P
(0,–ap2)
T
Notice that, since QP is parallel to ST , ∠QPB is equal to ∠ST P, so we will show that
∠ST P = ∠SPT .
Now S has coordinates (0, a), and T has coordinates (0,−ap 2 ), obtained by putting x = 0
in the equation of the tangent at P. Hence
SP 2 = (2ap − 0) 2 + (ap 2 − a) 2 = a 2 (p 2 + 1) 2