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EQUATION OF THE FIRST DEGREE, THE STRAIGHT LINE

The general form of an equation of the first degree in two variables is

ax + by + c = 0 .

Its locus is a straight line. Every straight line in the plane can be represented by a first degree equation in two variables. Every first degree equation in two variables is represented by a straight line in the plane.

Special forms of the equation of a straight line.

1] Point-slope form. The equation of a straight line that passes through a point P1(x1, y1) with a slope m is

y - y1 = m(x - x1)

2] Point-intercept form. The equation of a straight line having slope m and y intercept b is

y = mx + b

3] Two-point form. The equation of the straight line passing through points P1(x1, y1) and P2(x2, y2) is

4] Intercept form. The equation of the straight line whose x and y intercepts are a and b, respectively, is

5] Normal form. The normal form of the equation of a straight line is

x cos α + y sin α - p = 0

where α is the angle from the x-axis to the perpendicular from the origin to the line and p is the length of the perpendicular. See figure 1.

General form. The general form of the equation of a straight line is

ax + by + c = 0

where a, b and c are arbitrary constants. This form includes all other forms as special cases. For an equation in this form the slope is -a/b and the y intercept is -c/b. An equation in general form can be changed to normal form by dividing by

where the sign of the radical is taken opposite to that of c if c 0 and the same as that of b if c = 0. Thus the normal form of ax + by + c = 0 is

Perpendicular distance from a line to a point. The perpendicular distance d from the line ax + by + c = 0 to the point P1(x1, y1) is given by

where the sign of the radical is taken opposite to that of c if c 0 and the same as that of b if c = 0. The distance d is positive if P1 is on the opposite side of the line from the origin and negative if it is on the same side of the line as the origin.

Parallel lines.

1] The two lines y = m1x + b1 and y = m2x + b2 are parallel if m1 = m2.

2] The two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are parallel if a1 /a2 = b1 /b2.

Equation of a line through a point parallel to a given line. The equation of a line through the point P1(x1, y1) and parallel to line ax + by + c = 0 is

a(x - x1) + b(y - y1) = 0

Perpendicular lines.

1] The two lines y = m1x + b1 and y = m2x + b2 are perpendicular if m1 = -1/m2.

2] The two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are perpendicular if a1a2 + b1b2 = 0.

Equation of a line through a point perpendicular to a given line. The equation of a line through the point P1(x1, y1) and perpendicular to line ax + by + c = 0 is

b(x - x1) + a(y - y1) = 0

Intersecting lines.

1] Let a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 be the equations of two intersecting lines and λ an arbitrary constant. Then

(a1x + b1y + c1 ) + λ(a2x + b2y + c2 ) = 0

represents the system of lines through the point of intersection.

2] The three lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0, and a3x + b3y + c3 = 0 meet in a point if