Slope of a line
First, talk in intuitive terms about what is meant by slope. Give real-life examples of slope such as the slope of
the roof of a house, a road going up a hill, or a ladder leaning against a building. Explain that we can assign a
number that allows us to measure the steepness of a straight line. Also, say that the greater the absolute
value of this number, the steeper the line will be.
Slope of a non-vertical line L is the tangent of the angle θ, which the line L makes with the positive direction of
x-axis. In particular,
(a) Slope of a line parallel of x-axis is zero.
(b) Slope of a line parallel to y-axis is not defined.
(c) Slope of a line equally inclined to the axis is −1 or 1.
(d) Slope of a line making equal intercepts on the axis is −1.
(g) Slopes of two parallel (non-vertical) lines are equal. If m1, m2 are the slopes, then m1 = m2.
(h) If m1 and m2 be the slopes of two perpendicular lines (which are oblique), then m1m2 = - 1.
Straight-line equations, or "linear" equations, graph as straight lines, and have simple variables with no
exponents on them. If you see an equation with x and y, then you're dealing with a straight-line equation.
An equation of the form ax + by + c = 0 is called the general equation of a straight line, where x and y are
variable and a, b, c are constants.
Equation of a line parallel to X axis or Y - axis
(i) Equation of any line parallel to x-axis is y = b, b being the directed distance
of the line from the x-axis. In particular equation of x-axis is y = 0
(ii) Equation of any line parallel to y-axis is x = a, a being the directed distance
of the line from the y-axis. In particular equation of y-axis is x = 0.
(a) One point form
Equation of a line (non-vertical) through the point (x1, y1) and having
slope m is
y - y1 = m (x - x1).
(b) Two-point form
Equation of a line (non-vertical) through the points (x1, y1) and (x2, y2) is
(c) Slope-intercept form
Equation of a line (non-vertical) with slope m and cutting off an intercept c from the y-axis is
y = m x + c.
(d) Intercept form
Equation of a line (non-vertical) with slope m and cutting off intercepts a and b from the x-axis
and y-axis respectively is
Ex.1 Line intersects x axis at A (10, 0) and y-axis at B (0, 10). Find the equation of the line.
(1) x + y = 10
(2) x + y = 20
(3) x = - y
(4) None of these
Sol. As line intersects x-axis at A (10, 0)
⇒ length of intercept on x-axis, a = 10
Similarly length of intercept on y-axis, b = 10
∴ Using intercept form, equation of line is
or x + y = 10. Answer: (1)
Ex.2 Find the equation of the straight line passing through the point (- 2, - 3) and perpendicular to
the line through (- 2, 3) and (- 5, - 6).
(1) X + 2 Y + 8 = 0
(2) X + 3Y + 11 = 0
(3) X - 3Y = 7
(4) X + 3Y = 11
Sol. The slope of the line through (- 2, 3) and (- 5, - 6) is m = = 3
⇒ The slope m1 of the required line =
By point - slope form, Y + 3 =
⇒ X + 3Y + 11 = 0. Answer: (2)
Ex.3 Find the slope of the line passing through (- 3, 7) having Y-intercept - 2.
(1) - 5
(3) - 3
Sol. The line passes through the points (- 3, 7) and (0, - 2).
∴ Slope of the line = = - 3. Answer: (3)
Some Important Results
• Length of perpendicular from the point (x1, y1) to the line ax + by + c = 0 is
• Distance between parallel lines ax + by + c = 0 and ax + by + d = 0
• The angle between two lines y = m1x + b1 and y = m2x + b2 is given by
• The equation a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 represent the same line if
Three or more lines are said to be concurrent lines when all of them pass through a common point.