LINES 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 ° ≤ θ ≤ ° (Fig . ). We observe that lines parallel to x -axis, or coinciding with x -axis, have inclination of °.
The inclination of a vertical line (parallel to or coinciding with y -axis) is °. Definition 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 ° is not defined. The slope of a line is denoted by m .
Thus, m = tan θ , θ ≠ ° 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 coordinates 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( x , y ) and Q( x , y ) be two points on non-vertical line l whose inclination is θ . Obviously, x ≠ x , 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. . (i) and (ii).
Case When angle θ is acute: In Fig . (i), ∠ MPQ = θ . ... ( ) Therefore, slope of line l = m = tan θ .
But in ∆ MPQ, we have MQ tan θ MP ... ( ) Fig . Fig . (i) MATHEMATICS From equations ( ) and ( ), we have m Case II When angle θ is obtuse: