Coordinate Geometry Basics

1Introduction

This lecture converts conditions satisfied by a point $P(x,y)$P(x,y) in the plane into equations and interprets the geometric sets represented by those equations. The correspondence is developed for lines, circles, and distance conditions.

2Loci and equations

The locus of a point is the set of all its positions satisfying a given condition. If a condition on $P(x,y)$P(x,y) is equivalent to an equation $F(x,y)=0$F(x,y)=0, then

Both directions must be verified: the equation must neither introduce extraneous points nor omit required points.

3Distance between two points

For $A(x_1,y_1)$A(x_1,y_1) and $B(x_2,y_2)$B(x_2,y_2), the Pythagorean theorem gives the distance

This formula converts geometric distance conditions into equations.

4Lines

A nonvertical line whose $y$y-coordinate increases by $m$m whenever its $x$x-coordinate increases by one has equation $y=mx+n$y=mx+n. Here $m$m is the slope and $n$n is the $y$y-intercept. A vertical line cannot be expressed in this form and instead has equation $x=a$x=a. The general form covering both cases is

5Circles

A circle with center $C(a,b)$C(a,b) and radius $r>0$r>0 is the locus of points $P(x,y)$P(x,y) whose distance from $C$C is $r$r. Hence

Both sides are nonnegative, so squaring preserves equivalence and gives

Conversely, every point satisfying this equation has $CP=r$CP=r. Thus the equation represents precisely the entire circle.

6Completing the square

For example, completing the square in

gives

and therefore

The equation represents the circle with center $(1,-2)$(1,-2) and radius $3$3.

7Summary

• To determine a locus, verify the equivalence between the condition on $P(x,y)$P(x,y) and its equation.
• The general line equation $ax+by+c=0$ax+by+c=0 includes vertical lines.
• The standard circle equation is $(x-a{)}^2+(y-b{)}^2=r^2$(x-a)^2+(y-b)^2=r^2; completing the square identifies its center and radius.