Cartesian System of Rectangular Co-Ordinates, Inclination and Slope of a Line, Slope of a Line Joining Two Distinct Points

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Inclination and Slope of a Line

  • Look at the Fig. The line makes an angle or with the x-axis (measured in anticlockwise direction).

  • The inclination of the given line is represented by the measure of angle made by the line with the positive direction of x-axis (measured in anticlockwise direction)

  • In a special case when the line is parallel to x-axis or it coincides with the x-axis, the inclination of the line is defined to be .

The line AB makes an angle or π+α

The Line AB Makes an Angle or Π+Α

The line AB makes an angle or π+α

Again look at the pictures of two mountains given below. Here we notice that the mountain in Fig. (a) is more steep compaired to mountain in Fig. (b).

The pictures of two mountains

The Pictures of Two Mountains

The pictures of two mountains

  • How can we quantify this steepness? Here we say that the angle of inclination of mountain (a) is more than the angle of inclination of mountain (b) with the ground.

  • Try to see the difference between the ratios of the maximum height from the ground to the base in each case.

  • Naturally, you will find that the ratio in case (a) is more as compaired to the ratio in case (b). That means we are concerned with height and base and their ratio is linked with tangent of an angle, so mathematically this ratio or the tangent of the inclination is termed as slope. We define the slope as tangent of an angle.

  • The slope of a line is the tangent of the angle (say) which the line makes with the positive direction of x-axis. Generally, it is denoted by

Example:

Find the slope of a line which makes an angle of with the negative direction of x-axis.

Solution:

The angle of 30° slope line

The Angle of 30° Slope Line

The angle of 30° slope line

Here

slope of the line

Slope of a Line Joining Two Distinct Points

Let and be two distinct points. Draw a line through and and let the inclination of this line be . Let the point of intersection of a horizontal line through and a vertical line through be , then the coordinates of are as shown in the Fig.

Slope of a line joining two distinct points

Slope of a Line Joining Two Distinct Points

Slope of a line joining two distinct points

(A) In Fig.(a), angle of inclination is equal to (acute). Consequently.

(B) In Fig.(b), angle of inclination is obtuse, and since and are supplementary, consequently,

Hence in both the cases, the slope m of a line through and is given by

Example:

Find the slope of the line joining the points and .

Solution:

The slope of the line passing through the points and

Here,

Now substituting these values, we have slope

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