Example 7-2. Reflection property of the ellipse.
Consider the ellipse x
2
a^2
+y
2
b^2
= 1 having eccentricity e < 1 and foci at the points
F 1 , F 2 with coordinates (c,0) and (−c,0) respectively. For this ellipse b^2 =a^2 −c^2 and
c=ae. If Prepresents an arbitrary point (x 0 , y 0 )on the ellipse, then one can construct
the vector r 1 from Pto F 1 and also construct the vector r 2 from point P to F 2. The
magnitude of these vectors when summed gives
|r 1 |+|r 2 |= 2a (7 .5)
The vectors r 1 , r 2 and the ellipse are illustrated in the figure 7-2. If the ellipse is
mirrored, then a ray of light from the focus F 1 will reflect from an arbitrary point P
on the ellipse to the focus at F 2.
Figure 7-2. Light ray from one focus passes through other focus.
The position vector of a general point on the ellipse can be represented in the
parametric form
r =r (t) = acos tˆe 1 +bsin teˆ 2 , 0 ≤t≤ 2 π (7 .6)
A point P on the ellipse with coordinates (x 0 , y 0 )is described by equation (7.6) by
assigning the proper value for the parameter t. The proper value for the parameter
t, call it t 0 , is determined by solving the equations
x 0 =acos t and y 0 =bsin t
