Straight Line

 

Slope of a Line

A line in a coordinate plane forms two angles with the x-axis, which are supplementary. 

 The angle (say) θ made by the line l with positive direction of x-axis and measured anti clockwise is called the inclination of the line. 

Obviously 0° ≤ θ ≤ 180° (Fig 9.2). 

 We observe that lines parallel to x-axis, or coinciding with x-axis, have inclination of 0°. The inclination of a vertical line (parallel to or coinciding with y-axis) is 90°.

 Definition 1 

If θ is the inclination of a line l, then tan θ is called the slope or gradient of the line l. 

 The slope of a line whose inclination is 90° is not defined. 

 The slope of a line is denoted by m. Thus, m = tan θ, θ ≠ 90° 

 It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.

Slope of a line when co-ordinate of any two points on the line are given.

 We know that a line is completely determined when we are given two points on it. 

 Hence, we proceed to find the slope of a line in terms of the coordinates of two points on the line. 

 Let P(x1 , y1 ) and Q(x2 , y2 ) be two points on non-vertical line l whose inclination is θ. Obviously, x1 ≠ x2 , otherwise the line will become perpendicular to x-axis and its slope will not be defined. 

The inclination of the line l may be acute or obtuse. Let us take these two cases. 

 Draw perpendicular QR to x-axis and PM perpendicular to RQ as shown in Figs. 9.3 (i) and (ii).

 Case 1 

When angle θ is acute: 

 In Fig 9.3 (i), ∠MPQ = θ ............. (i) 

 Therefore, slope of line l = m = tan θ. MQ But in ∆MPQ, we have 

 tanθ = MQ/MP = y2 - y1/x2 - x1 ............... (ii)

m = y2 - y1/x2 - x1

Video Lecture Related to Slope of a Line when two points are given (For Acute Angle)

 Case II 

Video Lecture Related to Slope of a Line when two points are given (For Obtuse Angle)

 When angle θ is obtuse: In Fig 9.3 (ii), we have ∠MPQ = 180° – θ. 

 Therefore, θ = 180° – ∠MPQ. 

 Now, slope of the line l 

 m = tan θ = tan ( 180° – ∠MPQ) 

= – tan ∠MPQ = − QM/ PM  =− (y2 -  y1)/  (x1 -  x2) 

 Consequently, we see that in both the cases the slope m of the line through the points 

(x1, y1) and (x2,y2) is given by

m = y2 - y1/x2 - x1