Slope of a line is the tangent of the angle made by the straight line with positive x-axis in anticlockwise direction. It is denoted by m. If θ be the angle made by the straight line with positive x-axis in anticlockwise direction, then slope formula in terms of angle θ is given by,

Slope (m) = tanθ

### Slope of a line joining two given points

Let A (x1, y1) and B (x2, y2) be given two points. Join AB, and produce BA to meet x-axis at C so that XCB = θ

Then, slope of CB = slope of AB = m = tanθ.

From A and B, draw AM and BN perpendicular to OX. Again from A, draw AD perpendicular to BN.

DAB = XCB = θ
Run = AD = MN = ON – OM = x2 – x1
Rise = DB = NB – ND = NB – MA = y2 – y1

From the right angled triangle ADB,
Hence, the slope of a line passing through points (x1, y1) and B (x2, y2) is
This is called slope formula.

Note:
a.  The slope of the x-axis and the line parallel to x-axis are zero.
b.  The slope of the y-axis and the line parallel to y-axis are not defined.
c.  Two straight lines with equal slopes are parallel to each other.
d.  If the product of the slopes of two straight lines is -1, then the lines are perpendicular to each other.
e.  If slope of AB = slope of BC, then the points A, B and C are collinear.

### Workout Examples

Example 1: Find the slope of a line whose inclination is 60°.

Solution: Here,
Inclination (θ) = 60°
Slope (m) = tanθ
= tan60°
= √3

Slope of the line m = √3.

Example 2: Find the inclination of a line whose slope is 1.

Solution: Here,
Slope (m) = 1
i.e.       tanθ = 1
or,        tanθ = tan45° [ tan45° = 1]
θ = 45°

Inclination of the line is 45°.

Example 3: A straight line passes through the points (3, 2) and (8, 12). Find its slope.

Solution: Here,
The points are (3, 2) and (8, 12)
By the formula,

Slope of the line m = 2.

Example 4: If the slope of a line passing through the points P (2, - 2) and Q (4, a) is -1, find the value of a.

Solution: Here,
P(2, -2) and Q(4, a)
x1 = 2            x2 = 4
y1 = -2          y2 = a
Slope (m) = -1

Value of a is -4.

Example 5: If the points (k, 4), (-3, 2) and (3, 5) lie on the same straight line, find the value of k.

Solution: Let the given points are A(k, 4), B(-3, 2) and C(3, 5). As the points A, B and C lie on the same straight line,

Slope of AB = slope of BC

Value of k is 1.

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