Calculus: Analytic Geometry and Calculus, with Vectors

6.4 Hyperbolas 385

the hyperbola at Pl intersects the asymptotes of the hyperbola. fins.: The inter-

sections with the asymptotes y = (b/a)x and y = - (b/a)x are respectively

a2b ab2 ll a2b -ab2

Cbxl- ay,' bxl - ay,/' Cbxl+ ay,' bx, -I- ayi/

Remark: Since (bxi - ay,)(bx, + ay,) = b2xi - a2yi = a2b2, the denominators

are all positive or all negative.

15 Prove that if a line T, is tangent to a hyperbola at P,, then P, lies midway

between the points at which T, intersects the asymptotes of the hyperbola.

16 Find the area of the triangular region bounded by the tangent T, and the

asymptotes of the hyperbola of Problem 5. Ins.: ab.

17 Substantial information about the geometry of the hyperbola having the

standard equation

(1)

x2 y2

a2 b2

is concealed in Figure 6.494. The figure shows the auxiliary rectangle whose

vertices lie on the asymptotes, and also the asymptotes. As we know, the circle

Figure 6.494

with center at 0 and radius a2 + b2 intersects the x axis at the foci. The
smaller circles have radii a and b. The line RS is tangent to the circle of radius
b at the point (b,0) where this circle intersects the positive x axis. Upon the
basic part of the figure which has been described, we can heap more construction.
Let OT be a ray (or half-line) from the origin which makes with the positive x
axis an angle 0 for which 0 < 0 < a/2. Let fl be the point where the ray OT
intersects the circle having radius a, and let Q be the point where the tangent to
the circle at .4 intersects the x axis. Let B be the point where the ray OT inter-
sects the line RS. The point P(x,y) where the horizontal line through B intersects
the vertical line through Q lies on the hyperbola because
x=IOQI
= sec 4i =

(^1) , y_IRB-= tan sin 4,
a 15:11 cos 4, b IORI cos 45
and hence
x2 y2
= sect 4, - tan2 4, =
1 -sing - 1.
a2 b2 cost 0