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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.
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
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
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
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