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**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 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**

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 (x_{1}, y_{1}) and having

slope m is**y - y _{1} = m (x - x_{1}).**

**(b) Two-point form**

Equation of a line (non-vertical) through the points (x_{1}, y_{1}) and (x_{2}, y_{2}) 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 **

**(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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