Calculus: Analytic Geometry and Calculus, with Vectors

1.3 Lines and linear equations 15

the point (h,0) on the positive x axis as in Fig-
ure 1.292. Supposing that the coordinates of C
are (h + a, k), show that the coordinates of B
must be (h - a, -k). Find equations of the
lines containing the sides of the triangles. Write
the equations in the form y = mx + b and check
the answers by determining whether the coordi-
nates of the vertices satisfy the equations. I UB(k-a, -k)
16 The vertices P1(xl,yi), P2(x2,y2), P3(xa,ya)
of a triangle are unknown, but it is known that

C(h+a, k)

X

Figure 1.292

the mid-points of the sides P1P2, P2P3, and P3P1 are respectively (7, -1), (4,3), and
(1,1). Find the unknowns and check the results by drawing an appropriate
figure.
17 Formulate and solve a more general problem of which Problem 16 is a
special case.

1.3 Lines and linear equations; parallelism and perpendicularity

When -4, B, and C are constantsf for which -4 and B are not both 0, the
equation

(1.31) 'Ix+By+C=0

is a linear equation and we must both prove and remember that its graph
is a line. In case B 0 0, we can put the equation in the point-slope form

y - -

B\)=-B(x-0)

and see that the graph is the line L which passes through the point
(0, -C/B) and has slope -14/B. In case B = 0, we must have 14 $ 0,
and we can put the equation in the form

X C

The graph of this equation is the vertical line consisting of all those points

(x,y) for which x = -C/!4. This proves the

result. I -
The equation

(1.32) y = mx+b

can be put in the form y - b = m(x - 0), and
hence it is the equation of the lineL which passes

P(x, y)

Figure 1.33

through the point (0,b) and has slope m. The equation (1.32) is called

t The hypothesis that .4, B, and C are constants means merely that they are numbers that
are "given" or "selected" or "fixed" in some way. There is no implication that other num-
bers are unstable in the sense that they are moving around. We shall hear more about this
matter later.