What is slope of a straight line?
The tangent value of any trigonometric angle that a straight line makes with the positive direction of the xaxis in anticlockwise direction is called the slope or gradient of a straight line.
The angle of inclination of a line is the angle made by the line with the positive direction of the xaxis. It is usually measured from the positive xaxis in the anticlockwise direction.
The slope of the line is generally denoted by ‘m’. Thus, m = tan θ. The gradient or slope of a line (not parallel to the axis of y) is the trigonometrical tangent of the angle which the line makes with the positive direction of the xaxis. Thus, if a line makes an angle θ with the positive direction of the xaxis, then its slope will be tan θ. The slop of a line is positive or negative according as θ is acute or obtuse. Sine a line parallel to xaxis makes an angle of 0° with xaxis, therefore its slope is tan 0° = 0. A line parallel to yaxis that is i.e., perpendicular to xaxis makes an angle of 90° with xaxis, so its slope is tan \(\frac{π}{2}\) = infinity. Also the slope of a line equally inclined with axes is 1 or 1 as it makes 45° or 135° angle with xaxis.
In short, the slope of a line is the trigonometrical tangent of its inclination.
In the above figure the inclination of the lines MN and PQ are α and β respectively.
Solved examples to find the slope of a straight line:
1. Find the slope or gradient of a straight line whose inclination to the positive (+ve) direction of xaxis in anticlockwise direction is
(i) 30°
(ii) 0°
(iii) 45°
(iv) 135°
Solution:
(i) 30°
Slope or gradient = tan 30° = \(\frac{1}{√3}\)
(ii) 0°
Slope or gradient = tan 0° = 0
(iii) 45°
Slope or gradient = tan 45° = 1
(iv) 135°
Slope or gradient = tan 135° = cot 40° = 1
2. What can be said regarding a line if its Slope or gradient is
(i) (+ve)
(ii) Zero (0)
(iii) (ve)
Solution:
Let ∅ be the angle of inclination of the given straight line with the positive (+ve) direction of xaxis in anticlockwise direction. Then its Slope or gradient is given by m = tan ∅.
(i) Slope or gradient is positive (+ve)
⇒ m = tan ∅ > 0
⇒ ∅ lies between 0° and 90°
⇒ ∅ is an acute angle.
(ii) Slope or gradient is zero (0)
⇒ m = tan ∅ = 0
⇒ ∅ = 0°
⇒ either the line is xaxis or is parallel to xaxis.
(iii) Slope or gradient is negative (ve)
⇒ m = tan ∅ < 0
⇒ ∅ lies between 0° and 180°
⇒ ∅ is an obtuse angle.
11 and 12 Grade Math
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