388 Cones and conics
25 Let 0 < b < a. Prove that when X < b2, the ellipse having the equation
2 y2a2-X b2-X
has its foci at the points (± b2, 0). Prove that, when b2 < A < a2, the
hyperbola having the same equation has the same foci.
26 Let H be a given hyperbola. Describe an elementary geometric procedure
for locating the center, the axes, and the asymptotes of H. Hints: Take clues
from the hints of Problem 25 of Problems 6.39. Then use the fact that if the
equation of H has the standard form, then y = ±b when x is the length of a
diagonal of a square of which the sides have length a.
27 We have seen string mechanisms for construc-
tion of parabolas and ellipses, and we need not slight
hyperbolas. Let a red string run from a knot K so
that it passes below a tack Fl to an end tied to a pencil
point at P as in Figure 6.496. Let a white string run
from the same knot K so that it passes below the tack(^3) ` F, and then under a tack F2 to an end tied at P. Show
Figure 6.496 that if P is held in such a way that both strings are
kept taut, then P will trace a part of a branch of a
hyperbola as the knot K is pulled away from F,. Hint: Justify and use the fact
that if the red and white strings have lengths R and IV, then
IT,KI = R - W - IFiF2I - IFVI
at all times.
28 Sound travels with speed s, and a bullet travels with speed b from a gun
at (-h,0) to a target at (h,0) in an xy plane. Where in the plane can the boom
of the gun and the ping of the target be heard simultaneously? Ins.: At points
(x,y) for which
/(x + h)2 + Y2 1/(x - h)2 + y2 2h
J s + b
or-\/(X + h)2 + Y2= 1/(x -h)2 + y2 +2qh
where q = s/b and hence q = 1/M where M is the Mach number of the bullet.
In case M < 1 and q > 1, there are no such points because the length of one side
of a triangle cannot exceed the sum of the lengths of the other two sides. In
case M = 1 and q = 1, the required points are those for which y = 0 and x > h.
In case M > 1 (so the speed of the bullet is supersonic) and q < 1, the required
points lie on the right-hand branch of the hyperbola having the equation
x2 2
q2h2h2(1 - q2) = 1.29 We have seen that if S, is the set of points equidistant from a line and a
circle having its center on the line, then S, is the sum or union of two parabolas.
We have also seen that if S2 is the set of points equidistant from a circle and a
point inside the circle which is not the center of the circle, then S2 is one ellipse.
Complete the story by showing that if Ss is the set of points equidistant from a
circle and a point outside the circle, then S3 is half (one branch) of a hyperbola.