Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

142

ANALYTIC GEOMETRY

The student should compare these facts with the corresponding ones concerning a single ellipse and the circumscribed
rectangle.
EXERCISES
1. Find the coordinates of the foci, the lengths of the axes,
the slopes of the asymptotes, and the value of the eccentricity
for each of the hyperbolas:
(a) = 1; (c) y2 -x = 4;
(b) 52- 4y2+20= 0; (d) 3x - 2y2 + 6=0.
Draw an accurate figure in each case.
2. What are the equations of the hyperbolas conjugate to
the hyperbolas of Ex. 1?
3. Find the equation of the hyperbola whose vertices are in
the points (0, ~ 4) and whose eccentricity is 3.
Ans. 4 x2- 5y +80 = 0.
4. Find the equation of the hyperbola the extremities of
whose minor axis are in the points (~ 3, 0) and whose eccentricity is 4.
5. Prove that the sum of the squares of the reciprocals of
the eccentricities of the two conjugate hyperbolas
x2 y2 x2 y2
9 16 9 16
is equal to unity.
6. Prove the theorem of Ex. 5 for the general pair of conjugate hyperbolas.
7. Show that the foci of a pair of conjugate hyperbolas
lie on a circle.
9. Parametric Representation. It is possible to construct a
hyperbola, given its axes, AA' and BB', by a method much
like that of Ch. VII, ~ 10, for the ellipse.