Calculus: Analytic Geometry and Calculus, with Vectors

1.4 Distances, circles, and parabolas^31

20 A triangle has vertices Pj(xi,yi), P2(x2,y2),
P3(xa,ya). Prove analytically that 4 times the sum of
the squares of the lengths of the medians is equal to
3 times the sum of the squares of the lengths of the
sides.
21 Discover for yourself that a part of a parabola
can be drawn with the aid of the right triangle (or
rectangle), string, and pin mechanism shown in Figure
1.492. A string of length ED has one end fastened
to the triangle at E and has the other end fastened to
a pin at the focus F. A pencil point at P keeps the
string taut, so FP = DP, and traces a part of the

D

Figure 1.492

parabola as the base of the triangle is moved along the directrix. Such con-

structions are taboo in the classical ruler-and-compass geometry of Euclid, but

in analytic geometry we can recognize the existence of all kinds of machinery.

22 Supposing that p > 0, find and simplify the equation of the parabola

whose focus is at the origin and whose directrix is the line having the equation

y = -p. t1ns.: y = 2 (x2 - p2). Remark: If we set k = 1/2p, then. the

equation takes the form y = k(x2 - 1/4k2). The parabolas obtained by taking

different values of p or k constitute a family of confocal parabolas; concentric

circles have the same center and confocal parabolas have the same focus.

23 Supposing that p < 0, find and simplify the equation of the parabola

whose focus is at the origin and whose directrix is the line having the equation

y= -p.

24 Supposing that p > 0, find and simplify the equation of the parabola

whose focus is at the origin and whose directrix is the line having the equation

x = P. .4ns.: x = Zp (y2 - p2).

25 Find the equation of the parabola whose focus is the point (12,0) and whose

directrix is the line having the equation x = -12. .Ans.: x = y2/48.

26 Supposing that k # 0, use the distance formula to obtain the equation

satisfied by the coordinates (x,y) of points P equidistant from the point F(xo,

yo + 1/4k) and the line L having the equation y = yo - 1/4k. Outline offsolu-

tion: A point P(x,y) lies on the parabola if and only if FP = DP, where D is the

point (x, yo - 1/4k). Writing FP and DP in terms of coordinates gives an

equation which reduces to (1.473).

27 Supposing that h > 0 and X > 1, find and simplify the equation satisfied

by the coordinates of the points P(x,y) whose distances from the point A(-h,0)

are X times their distances from the point B(h,0). Ans.:

\x X2

- 1h\2+

Y^2

X221 1 h)2.

28 Stillsupposing`that h > 0 and X > 1, show that the graph of the answer

to Problem 27 is a circle having its center at a point Po on the x axis. Find the

x coordinates of the points Pl and P2 where the circle intersects the x axis. Ans.:

See Figure 1.191, which displays the x coordinates of the points and shows their

correct positions relative to A, 0, B and to each other.