Calculus: Analytic Geometry and Calculus, with Vectors

1.5 Equations, statements, and graphs 35

equation of the line through Po perpendicular to L and show that it is equivalent
to the second of the two equations. Solve these equations to find that the
coordinates xl, yl of the foot P, of the perpendicular from Po to L are

(2)

Show that

B2xo - .4Byo - 14C f12yo - f4Bxo - BC

xl = Z2 + B2 yl = 42 + B2

(3) xi - xo = 72 +B2(Axo + Byo + C),

Yi - yo =J2+

B2

(Axo + Byo + C).

Finally, use the fact that the distance d from Po to L is the distance from Po to

P, to obtain the formula

(4) d=(Axo+Byo+CI

. /42 + B2

1.5 Equations, statements, and graphs The equation y = x + 2

can be regarded as a statement that is true for some pairs of values of x

and y, for example, x = 3, y = 5, and is false for some other pairs of

values of x and y, for example, x = 7, y = 7. A similar remark applies

to each of the equations x2 + y2 = 4, Ox + Oy = 1, and Ox + Oy = 0,

and to each of the inequalities 0 < x < 1, y < x, and x2 + y2 < 1.

Each is a statement that is true for some (or none or all) pairs of values

of x and y and is false for the remaining ones. The graph of such a state-

ment is the set or collection of points P(x,y) whose coordinates are pairs

of values of x and y for which the statement is true. For example, the

graph of the statement (or equation) y = x is a line L. We can always

know that there is a substantial difference between an equation (or state-

ment) and its graph (a point set). Hence, we may be carrying abbrevi-

ation of language a bit too far when we sometimes follow the old and mis-

leading custom of referring to "the line y = x" instead of to "the line

having the equation y = x." In any case, we should think about this

matter enough to know that we are introducing analytic geometry and

hopefully trying to make sense out of nonsense if we receive a mysterious

order to "find the part of y = x in x2 + y2 = 1" and proceed to find the

length of the part of the line having the equation y = x which lies inside

the circle having the equation x2 + y2 = 1.t

Most of the graphs that appear in our work are graphs of equations.

However, graphs of inequalities can be important, and we look at some

simple examples. The graph of the inequality xy > 0 consists of those

points P(x,y) in the first quadrant (where x and y are both positive)

t Persons who start picking up clear ideas about these things may even enjoy studying

statements and sets in mathematical logic and elsewhere.