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Slope of a line
First, talk in intuitive terms about what is meant by slope. Give reallife 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 nonvertical line L is the tangent of the angle θ, which the line L makes with the positive direction of
xaxis. In particular,
(a) Slope of a line parallel of xaxis is zero.
(b) Slope of a line parallel to yaxis 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 (nonvertical) lines are equal. If m_{1}, m_{2} are the slopes, then m_{1} = m_{2}.
(h) If m_{1} and m_{2} be the slopes of two perpendicular lines (which are oblique), then m_{1}m_{2} =  1.
Straight line
Straightline 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 straightline 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 xaxis is y = b, b being the directed distance
of the line from the xaxis. In particular equation of xaxis is y = 0
(ii) Equation of any line parallel to yaxis is x = a, a being the directed distance
of the line from the yaxis. In particular equation of yaxis is x = 0.
(a) One point form
Equation of a line (nonvertical) through the point (x_{1}, y_{1}) and having
slope m is
y  y_{1} = m (x  x_{1}).
(b) Twopoint form
Equation of a line (nonvertical) through the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is
(c) Slopeintercept form
Equation of a line (nonvertical) with slope m and cutting off an intercept c from the yaxis is
y = m x + c.
(d) Intercept form
Equation of a line (nonvertical) with slope m and cutting off intercepts a and b from the xaxis
and yaxis respectively is
Ex.1 Line intersects x axis at A (10, 0) and yaxis 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 xaxis at A (10, 0)
⇒ length of intercept on xaxis, a = 10
Similarly length of intercept on yaxis, 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 Yintercept  2.
(1)  5
(2) 2
(3)  3
(4)
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 (x_{1}, y_{1}) 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 = m_{1}x + b_{1} and y = m_{2}x + b_{2} is given by
• The equation a_{1}x + b_{1}y + c_{1} = 0 and a_{2}x + b_{2}y + c_{2} = 0 represent the same line if
Concurrent Lines:
Three or more lines are said to be concurrent lines when all of them pass through a common point.
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