Calculus: Analytic Geometry and Calculus, with Vectors

6.4 Hyperbolas 383

4 The two vertices and one focus of a hyperbola are given. Describe a pro-

cedure by which it is possible to draw the asymptotes without using equations.

5 For basic studies of hyperbolas, we place the hyperbolasupon coordinate

systems in such a way that their equations have the standard form

x2 y2

62 T2 = 1.

Supposing P,(x,,yi) is, asin Figure 6.491, a point on the hyperbolathat is not a

Figure 6.491

vertex, find the equation of the line T, tangent to the hyperbola at Pl. Ans.:

xix YiY= 1.

a2 - b2

(^6) Find the coordinates of the points at which the tangent T, of Problem
5 intersects the coordinate axes. Ins.:
Cx,, o), (o, -y,}
(^7) Find the coordinates of the points where the tangent T, of Problem 5
intersects the lines through the foci perpendicular to the x axis. fins.:
(-ae,

• Yi C1 +

eai//'

(ae, y,Cl eat)

(^8) Find the coordinates of the points where the tangent T, of Problem 5
intersects the directrices of the hyperbola. .4ns.:
a h2 12
C e' YiC1 + ae/)' Ce' Y1(1 ae/l
9 Let the line T, tangent to a hyperbola at P, intersect the directrix at Q,
and let F be the focus corresponding to the directrix. With the aid of Problem 8
and the fact that b2 = a2(e2- 1), prove that the line FQ, is perpendicular to the
line FP1.