Chapter 4: Geometry

(a) The area of the ellipse is
¾
Ô Ô
½ .
¾
¾
(b) The ellipse has the parametric representation x´Øµ Ó״صv ½ ·×Ҵصv ¾ .
(c) The rectangle with vertices ´¦v ½ ¦v ¾ µ is tangent to the ellipse.
3. A rational parametric representation for is given by
½ ؾ
½·Ø ¾
4. The polar equation for in the usual polar coordinate system is
Ö
¾Ø
.
½·Ø ¾
Ô
¾ ×Ò ¾ · ¾ Ó× ¾ (4.6.12)
With respect to a coordinate system with origin at a focus, the equation is
Ö
Ð
½ ¦ Ó× (4.6.13)
where Ð ¾ is half the latus rectum. (Use the · sign for the focus with
positive Ü-coordinate and the sign for the focus with negative Ü-coordinate.)
5. Let È be any point of . The sum of the distances È and È ¼ is constant
and equal to ¾.
6. Let È be any point of . Then the rays È and È ¼ make the same angle
with the tangent to at È . Thus any light ray originating at and re ected
in the ellipse will go through ¼ .
7. Let Ì be any line tangent to . The product of the distances from and ¼ to
Ì is constant and equals ¾ .
8. Lahire’s theorem: Let and ¼ be x ed lines in the plane, and consider a
third moving line on which three points È , È ¼ , and È ¼¼ are marked. If we
constrain È to lie in and È ¼ to lie in ¼ , then È ¼¼ describes an ellipse.
4.6.4 ADDITIONAL PROPERTIES OF HYPERBOLAS
Let be the hyperbola with equation Ü ¾ ¾ Ý ¾ ¾ ½, and let
¼ ´¦ Ô ¾ · ¾ ¼µ (4.6.14)
be its foci (see Figure 4.17). The conjugate hyperbola of is the hyperbola ¼ with
equation Ü ¾ ¾ · Ý ¾ ¾ ½. It has the same asymptotes as , the same axes
(transverse and conjugate axes being interchanged), and its eccentricity ¼ is related
to that of by ¼ ¾
· ¾ ½.
© 2003 by CRC Press LLC