Slope of a line | Slope Formula

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

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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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