Calculus: Analytic Geometry and Calculus, with Vectors

360 Cones and conics

that the two points Pi(xi,kxi), P2(xs,kz2) on the graph of y = kx2 are end points
of a focal chord if and only if (2kxi)(2kx2) _ -1 and hence if and only if the
tangents to the parabola at P, and P2 are perpendicular.
17 With or without the aid of the results of the preceding problems, show
that two different tangents to a parabola intersect on the directrix if and only if
the tangents are perpendicular and hence if and only if the points of tangency
are ends of a focal chord.
18 Prove that the center of a focal chord of a parabola is equidistant from the
directrix and the ends of the chord.
19 Two equilateral triangles in E2 are similar in the sense that one can be
transformed into the other by a translation, a rotation, and a change of scale
Show that the same is true of two parabolas in E2. Hznt: Suppose that the to
given parabolas are translated and rotated so that their equations become
y = k1x2 and y = k2x2, where ki and k2 are positive constants. Show that if in
the first equation we change scale by replacing x and y by Ax and Ay, we obtain
y = (Akl)x2
20 Let k be a positive constant. For each positive number a, let F(a) be
the y coordinate of the center of the circle tangent to the graph of y = kx2 at
the points for which x = a and x = -a. Find F(a) and lim F(a). Ans.:

ka2 + 1/2k and 1/2k.

21 Let k be a positive constant. For each positive number a, let (G(a),

H(a)) be the center of the circle which is tangent to the graph of y = kx2 at the

point (a,ka2) and which contains (or passes through) the origin. Show that

and

G(a) = 2 ka2 + Zk H(a) =-k2a3

lim G(a) = 1/2k, lim H(a) = 0.

22 Sketch a graph of the parabola having the equation y = x2 and then

sketch several circles which have centers on the positive y axis and are tangent to

the x axis at the origin. Observe that sufficiently big circles in this family inter-

sect the parabola at points different from the origin and that small circles leave

us in doubt. Supposing that k > 0, investigate this matter for the parabola

having the equation y = kx2. fins.: The circle with center at (O,a) and radius

a intersects the parabola only at the origin (and is elsewhere above or "inside"

the parabola) if and only if a S 1/2k. Thus the biggest one of these circles

which lies completely on or inside the parabola has radius equal to the distance

from the focus to the directrix of the parabola.

23 Study the set S which contains a point P(x,y) if and only if the point is

Figure 6.195

equidistant from the x axis and the circle with center at

the origin and radius a. Solution: This problem is interest-

ing because S contains some points inside the circle as

well as some points on and some points outside the circle;

see Figure 6.195. Whether a point P(x,y) lies inside or

on or outside the circle, it will be in the set S iff (if and

only if)

(1) I x2 + y2 - al = Iyi