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If the lines l1 and l2 are perpendicular
m1 m2 = – 1
Example Find the slope of the lines:
(a) Passing through the points (3, – 2) and (7, – 2),
(b) Making inclination of 60° with the positive direction of x-axis.
Solution (a) The slope of the line through the points (3, – 2) and (7, – 2) is

(b) Here inclination of the line = 60°. Therefore, slope of the line is
m = tan 60° =√
10.2.3 Angle between two lines When we think about more than one line in a plane,
then we find that these lines are either intersecting or parallel. Here we will discuss the
angle between two lines in terms of their slopes

Example Line through the points (–2, 6) and (4, 8) is perpendicular to the line
through the points (8, 12) and (x, 24). Find the value of x.
Solution Slope of the line through the points (– 2, 6) and (4, 8) is

Slope of the line through the points (8, 12) and (x, 24) is

Since two lines are perpendicular, m1 m2 = –1, which gives

10.2.4 Collinearity of three points if A, B and C are three points in the XY-plane, then they will
lie on a line, i.e., three points are collinear ,if and only if slope of AB = slope of BC.
Example Three points P (h, k), Q (x1, y1) and R (x2, y2) lie on a line. Show that
(h – x1) (y2 – y1) = (k – y1) (x2 – x1).
Solution Since points P, Q and R are collinear, we have
Slope of PQ = Slope of QR, i.e.,

10.3 Various Forms of the Equation of a Line
10.3.1 Horizontal and vertical lines If a horizontal line L is at a distance a from the
x-axis then ordinate of every point lying on the line is either a or – a
Therefore, equation of the line L is either y = a or y = – a.
Example Find the equations of the line parallel to axes and passing through (– 2, 3).
Solution The y-coordinate of every point on the line parallel to x-axis is 3, therefore, equation
of the line parallel to x-axis and passing through (– 2, 3) is y = 3. Similarly, equation of the line
parallel to y-axis and passing through (– 2, 3) is x = – 2.
10.3.2 Point-slope form
Suppose that P0 (x0, y0) is a fixed point on a non-vertical line L, whose slope is m. Let P (x, y) be