Calculus: Analytic Geometry and Calculus, with Vectors

12 Analytic geometry in two dimensions

slope equation (or formula)

(1.27) y - y1 = m(x - xi).

Proof of this theorem is very simple. When x = xi, the point P(x,y)

lies on the line if and only if y = yi and hence if and only if (1.27) holds.

When x x1, the point P(x,y) lies on the line if and only if (1.252) holds

and hence if and only if (1.27) holds. This proves the theorem. Formula

(1.27) is known as the point-slope formula, and it must be permanently

remembered.

In accordance with general terminology which we shall introduce in

Section 1.5, (1.27) is an equation of the line which passes through the point

(xl,yi) and has slope m. Moreover, the line is the graph of the equation.

When we are required to obtain an equation of the line which passes

through the point (2, --) and has slope 3, we put x1 = , y1

m = 3, and write immediately

y+ 3(x-2).

Sometimes, but not always, it is desirable to put this equation in one of

the forms

y=3x-, 12x-4y-7=0,

and we tolerate the custom which allows any one of the three equations

to be called "the" equation of the line. Conversely, when we are required

to draw or sketch the graph of the equation

y + j = 3 (x - 201

we observe that the equation has the point-slope form with x1 = 2,

y1 = -I, m = 3 and then immediately draw the line through the point

(J,--4L) having slope 3. Problems at the end of this section provide

opportunities for practice in the art of using these ideas quickly, neatly,

and correctly.

When, as sometimes happens, we want to find an equation of the line

which passes through two given points PL(x1,y1) and P2(x2,y2) for which

x2 76 x1, we determine the slope m of the line from the formula

(1.28) in =y2 - y1 or y1 - y2

X2 - x1 x1 - x2

and then use the point-slope formula. For example, the slope of the line

passing through the points (3,-4) and (-2,1) is -5/5 or -1, and the

equation of the line through these points is y + 4 = -1(x - 3).

Problems 1.29

(^1) With Figure 1.24 out of sight, plot the points 4(3,2), B(-4,-2), C(4,0),
D(0,-2), and F(-2,3). If correct results are not obtained, read the explanations
of the text and try again.