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

THE HYPERBOLA

149

where c is an arbitrary constant, not zero, all have the same
asymptotes.
3. How many hyperbolas are there with the lines
3x2 -16y = 0
as asymptotes? Find an equation which represents them all.
Ans. 3x2 -16y2=c, c* 0.
4. What is the equation of all the rectangular hyperbolas
with the axes of coordinates as axes?
5. A hyperbola with the lines 4 x2 - y = 0 as asymptotes
goes through the point (1, 1). What is its equation?
Ans. 4x —y2=3.
6. The asymptotes of a hyperbola go through the origin
and have slopes ~ 2. The hyperbola goes through the point
(1, 3). Find its equation. Ans. 4 x - y2 - 5.
7. The two hyperbolas of Exs. 5 and 6 have the same
asymptotes, but lie in the opposite pairs of regions into which
the plane is divided by the asymptotes. Show that the sum
of the squares of the reciprocals of their eccentricities equals
unity.
8. Prove that of the hyperbolas of Ex. 2 those for which
c is positive are all similar, and that this is true also of those
for which c is negative. If e is the common value of the eccentricity of the hyperbolas of the first set and e' is that of
the hyperbolas of the second set, show that
1+1
~~(1) ~e2 e/2
9. Plove that the relation (1) is valid for the eccentricities
of any two hyperbolas which have the same asymptotes but
lie in the opposite regions between the asymptotes.
10. Show that two hyperbolas which are related as those
described in the previous exercise have the same eccentricity
if and only if they are rectangular hyperbolas.