6.3 Ellipses 377Show that(7)
Show that(8)IvI = b2 + (a2 - b2) sin2 t.-X a --b2 sin t[a + X a2 - b2 cos t].
Letting )k be the angle which the vector FP makes with the forward tangent v
at P, show with the aid of (6), (7), and (8) that
(9) TIP--v _ -X a - b2 sin t
cos k -
IFkPIIvI 1/b2 + (a2 - b2) sin2 t
Show that y = b sin t and that multiplying the numerator and denominator of
the last member of (9) by b gives the formula(10) cos ok =-X a2 - b2 y
1/b* + (a2 - b2)y2
Remark: These remarkable formulas yield the famous reflection property of
ellipses. Since X = 1 when k = 1 and X = -1 when k = 2, the numbers
cos 41 and cos y52 in (10) differ only in sign. This implies that the vectors
F P and F2P make supplementary angles with the forward tangent v to the ellipse
at P and hence that the lines F1P and F2P make equal angles with the normal to
the ellipse at P. This implies that if light or something else goes in a line from
F2 and is reflected from the ellipse in such a way that the angle 9, of reflection
is equal to the angle B, of incidence, then its path leads to F2. Moreover, because
of the string property of the ellipse, radiation leaving F1 at the same time but in
different directions will arrive simul-
taneously (or in phase) at F2-
21 The reflection property of ellipses
is a consequence of another interesting
geometric property of ellipses. Let the
line T of Figure 6.396 be tangent at P
to the ellipse having foci at F1 and F2.
Let H1 and H2 be the reflections in T
of F1 and F2; this means that T is the
perpendicular bisector of the line
segments F1H1 and F2H2. Then, as
indicated by the figure, the line seg-
ments F1H2 and F2H1 intersect at P.
To prove this fact, let 4 be any point
on T different from P and let B be theFigure 6.396point at which the line segment F1A intersects the ellipse. Then (why ?)
(1) IF-,BI + IBFzi < IF1JI + I4F2I
so (why?)
(2) IF1PI + IPFzi < IF1AI + I F I
and (why?)
(3) IF1PI + IPH I < IF -,,41 + 1,4H21.