368 Cones and con ics
lies on the hyperbola, and we start by
Y locating it in Figure 6.291. We then
Ll
Menter the wholesale sketching business
3 and sketch the line M containing 0 and
r ---- - - 7f F1( 2k, 2k) F, the circle C with center at 0 which con-
I C j) tains Y, and the square S whose sides are
tangent to C at the points where C inter-/ sects the coordinate axes. The points
x Fl( 2k, 2k) andF2(- 2k, - 2k)
I where the square S meets the line M are
Ix+y= the foci of the hyperbola. The line L,
joining the points where the circle and
/ F2 square meet the positive x and y axes is
Figure 6.291 one directrix of the hyperbola. The line
L2 joining the points where the circle and
square meet the negative x and y axes is the other directrix of the hyperbola.
Finally, we complete Figure 6.291 by sketching the hyperbola. As we know, the
coordinate axes are asymptotes of the hyperbola The origin is the center (the
center of symmetry) of the hyperbola. The line M is a line of symmetry; it is
called the transverse axis of the hyperbola. The line through 0 perpendicular to
M is another line of symmetry; it is called the conjugate axis of the hyperbola.
The particular hyperbola we have been studying is called a rectangular hyperbola
because its asymptotes are at right angles to each other.
6 Information theory teaches that results like those of Problem 5, which
depend upon considerable calculation, should be checked when it is relatively
easy to do so. Letting F be the point /2k), letting L be the line having
the equation x + y - = 0, letting D be the foot of the perpendicular from
P to L, and letting e = -V/2-, simplify the intrinsic equation IFP12 = e2JDPI2 to
obtain the coordinate equation xy = k. Remark: We are seldom required to
find the distance from a point to a line which is not parallel to a coordinate axis,
and it is helpful to be able to find Theorem 1.48, which enables us to obtain the
result very quickly.
7 Find the set of numbers k such that there exists at least one point (x,y)
whose coordinates satisfy the equation y = kx and the equation
(a) xt + y2 = 1 (b) xy = 1 (c) xy = -1
x2 y2 x2 y2
(d) 9+4=1 (e) 9-4=1
In each case, sketch a figure which shows the geometric significance of the result.
8 Tell why a circle is not an ellipse. Remark: The answer must be based
upon definitions and not upon intuitions of the untutored.
9 When .1 0 0 and B > 0, the graph of the equation
y==e1x2+B
is a central conic (circle or ellipse or hyperbola but not a parabola or a degenerate
conic) having its center at the origin. Show that if m and b are constants such
that the line having the equation
y=mx+b