Calculus: Analytic Geometry and Calculus, with Vectors

384 Cones and conics

10 Sketch a figure somewhat like Figure 6.491 and observe that L1
from a focus F perpendicular to the line T1 tangent to the hyperbola at P, and
the line OP1 from the center of the world to P, seem to intersect at a point on the
directrix. Prove that each of these lines does intersect the directrix at the point

(a/e, ayl/exi)
11 Figure 6.492 illustrates the fact that the line 71P1 from a vertex of a
hyperbola to a point P1 on the hyperbola is parallel to the line OE from the center

Figure 6.492

of the hyperbola to the point E where the tangent T, at P, intersects the tangent
z
at the other vertex. Show that the coordinates of E are (a,
91

( I-1))and

prove the fact.
12 Figure 6.493 shows a part of the hyperbola having, as usual, the standard

Figure 6.493

(^13) Using the notation and results of
Problem 12, show that the equation of
the tangent to the hyperbola at G is
ex - y = a. Remark: This shows that
the tangent to the hyperbola at G inter-
sects the x axis (the transverse axis of the
hyperbola) where a directrix does and
intersects the y axis (the conjugate axis of the hyperbola)at a point on the circle
which has its center at the center of the hyperbola and includes the vertices of
the hyperbola.
(^14) Let Pj(xl,y,) be a point on a hyperbola having the standard equation
x2/a2 - y2/b2 = 1. Find the coordinates of the points at which the tangent T1 to
equation
(1)
xz

y2

a2 T2=1.

Let G be the point in the first quadrant

where the hyperbola is intersected by the

line through the focus F parallel to the

directrix. Prove that, as the figure

shows, the coordinates of G are (ae, b2/a).

Remark: A relation among a, b, and e is

F(-,0) X needed.