Calculus: Analytic Geometry and Calculus, with Vectors

1.4 Distances, circles, and parabolas 29

having a right angle at C, and hence that the lines

BC and C.1 must be perpendicular. Calculate

the slopes of these lines and verify the perpen-

dicularity. Make everything check.

2 Figure 1.491 illustrates the familiar fact

that, when P1(xl,yl) andP2(x2,y2) are two distinct

points in a plane, the set of points P(x,y) equi-

distant from P1 and P2 is the perpendicular bi-

sectorL of the line segmentP1P2. Equate expres-

sions for the distance PP1 and PP2 and simplify the

of L in the form

P2 (X2, Y2)

Figure 1.491

result to obtain the equation

r xl + x21 r Yl + Y21

(X2-XI) x-^2 )+(Y2-yl)1\Y- 2 /I =0.

Then show that this line passes through the mid-point

P(xl + x2 Y1 + Y2

2 2 )

of the segment P1P2 and is perpendicular to the segment.

3 Sketch a figure showing the triangle having vertices at the three given

points and then calculate distances to determine whether the triangle is isosceles

(that is, has two sides of equal length):

(a) (1,0), (8,2), (3,-7) (b) (1,4), (6,-1), (7,6)

(c) (O,a), (a,O), (b,b) (d) (a,a), (-a,-a), (b,-b)

4 Find the length of the part of the x axis which lies inside the triangle

having vertices at the points (-3,-1), (5,1), and (1,5). Use a figure to deter-

mine whether the answer is reasonable.

5 Find the point on the x axis equidistant from the two points P1(-2,-1)

and P2(4,3) in two different ways. First, find the equation of the perpendicu-

lar bisector of the line segment P1P2 and find the point where this bisector inter-

sects the x axis. Then, with the aid of the distance formula, determine x so

that the distance from (x,0) to Pl is equal to the distance from (x,0) to P2-

6 Find the center and radius of the circle having the equation

(x - 1) (x - 5) + (y + 4) (y - 2) =0.

Show that the center is the mid-point of the line segment joining the points
J(1,2) and B(5,-4).
7 Find the center and radius of the circle having the equation

(x-xl)(x-x2)+(Y-Yl)(Y-Y2)=0.

8 Show that the equation of the circle C with center at Po(xo,yo)

a can be put in the form

I

and radius

(x - Xo)(X - xo) + (y - yo)(y - yo) = a2