SLOPE OF THE LINE

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The tangent of angle of the line with the x – axis in anticlockwise direction is called slope of the line.

It is denoted by m.

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Slope of the line AB,

{m_{AB}} = \tan \theta  

Example 1: Find the slope of a line which makes an angle of 60^\circ with the positive direction of y – axis.
Solution 1:
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Angle with x – axis in positive direction = 150^\circ

\therefore {m_{AB}} = \tan 150^\circ

 = \,\tan \,\left( {180^\circ  - 30^\circ } \right)

 =  - \tan 30^\circ

 = \, - \frac{1}{{\sqrt 3 }}

SLOPE OF A LINE IN TWO POINT FORM

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Let A and B be two points on the line whose coordinates are \left( {{x_1},\,{y_1}} \right) and \left( {{x_2},\,{y_2}} \right) respectively.

Draw AP \bot \,OX

BQ \bot OX

AM \bot BQ

AM = PQ = OQ – OP = x2 – x1

BM = BQ – QM = BQ – AP =y2 – y1

{\rm{In }}\Delta ABM,\,\,\tan \theta  = \frac{{BM}}{{AM}}

{m_{AB}} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

 

Example 2: Find the slope of a line passing through the points A(5, – 2) and B(–3, 4).

Solution 2:

{m_{AB}} = \,\frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

 = \,\frac{{4 - \left( { - 2} \right)}}{{ - 3 - 5}}

 = \,\frac{6}{{ - 8}}

 = \, - \frac{3}{4}

 

 

NOTE:

1. Slope of a line parallel to x – axis is zero.

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The angle of line AB with x – axis zero is: {m_{AB}} = \tan 0 = 0 .

2. Slope of a line parallel to y – axis is \infty .

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The angle of line PQ with axis is \frac{\pi }{2} .

\therefore The range of slope is \left( { - \infty ,\,\infty } \right) .

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