# 4 Lines

The (cartesian) equation of a straight line is linear in the coordinates x and y, that is, of the form ax+by+c=0.

The slope of this line is -a/b, the intersection with the x-axis (or x-intercept) is x=-c/a, and the intersection with the y-axis (or y-intercept) is y=-c/b. If a=0 the line is parallel to the x-axis, and if b=0 the line is parallel to the y-axis.

(In an oblique coordinate system everything in the preceding paragraph remains true, except for the value of the slope.)

When a +b =1 and c 0 in the equation ax+by+c=0, the equation is said to be in normal form. In this case c is the distance of the line to the origin, and = arcsin a = arccos b

is the angle that the perpendicular dropped to the line from the origin makes with the positive x-axis (Figure 1). Figure 1: The normal form of L is x cos + y sin =p.

To reduce an arbitrary equation ax+by+c=0 to normal form, divide by ± , where the sign of the radical is chosen opposite the sign of c when c 0 and the same as the sign of b when c=0.

### Lines with prescribed properties

Line of slope m intersecting the x-axis at x=x :

y=m(x-x )

Line of slope m intersecting the y-axis at y=y :

y=mx+y Line intersecting the x-axis at x=x and the y-axis at y=y :

x/x +y/y =1

(This formula remains true in oblique coordinates.)

Line of slope m going though (x ,y ):

y-y =m(x-x )

Line going through points (x ,y ) and (x ,y ): (These formulas remain true in oblique coordinates.)

Slope of line going through points (x ,y ) and (x ,y ):

(y -y )/(x -x )

Line going through points with polar coordinates (r ,  ) and (r ,  ):

r(r sin( -  )-r sin( -  ))= r r sin(  -  ).

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Silvio Levy
Wed Oct 4 16:41:25 PDT 1995

This document is excerpted from the 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press). Unauthorized duplication is forbidden.