Physics and Mathematics
Slope or Gradient of a Straight Line
1. Statement / Concept Overview
The slope (or gradient) of a straight line is a numerical measure that describes the steepness of the line and the direction in which it rises or falls as we move from left to right along the x-axis.
In simple words, slope tells us how much y changes when x changes by one unit.
2. Clear Explanation and Mathematical Derivation
Geometrical Interpretation
Consider a straight line making an angle [\theta] with the positive direction of the x-axis, measured anticlockwise.
The slope [m] of the line is defined as:
[m = \tan \theta]
This definition directly connects coordinate geometry with trigonometry.
Slope Using Two Points
Let a straight line pass through two points
[A(x₁, y₁)] and [B(x₂, y₂)].
Change in x-coordinate (horizontal change):
[\Delta x = x₂ − x₁]
Change in y-coordinate (vertical change):
[\Delta y = y₂ − y₁]
Hence, slope [m] is:
[m = \dfrac{\text{Change in y}}{\text{Change in x}} ][= \dfrac{y₂ − y₁}{x₂ − x₁}], provided [x₂ \neq x₁]
Special Geometrical Cases
- Horizontal Line
Here, [y₂ = y₁]
[m = \dfrac{0}{x₂ − x₁} = 0] - Vertical Line
Here, [x₂ = x₁]
[m = \dfrac{y₂ − y₁}{0}] → Not defined
3. Key Features of Slope
- Slope determines the inclination of a line.
- Slope indicates the direction of motion along the line.
- A single number [m] completely describes the tilt of a straight line.
- Parallel lines always have equal slopes.
- Perpendicular lines have slopes that are negative reciprocals of each other.
4. Important Formulas to Remember
| Situation | Formula for Slope |
|---|---|
| Line makes angle [\theta] with x-axis | [m = \tan \theta] |
| Line through [ (x₁, y₁), (x₂, y₂) ] | [m = \dfrac{y₂ − y₁}{x₂ − x₁}] |
| Horizontal line | [m = 0] |
| Vertical line | Slope not defined |
| Parallel lines | [m₁ = m₂] |
| Perpendicular lines | [m₁ m₂ = −1] |
5. Conceptual Questions with Detailed Solutions
1. What does a positive slope indicate?
A positive slope means that as x increases, y also increases. Geometrically, the line rises upward from left to right, making an acute angle with the positive x-axis.
2. What does a negative slope represent?
A negative slope indicates that as x increases, y decreases. The line falls downward from left to right and makes an obtuse angle with the positive x-axis.
3. Why is the slope of a vertical line not defined?
For a vertical line, the change in x-coordinate is zero. Since slope is defined as [\dfrac{\Delta y}{\Delta x}], division by zero is not possible, so the slope is not defined.
4. Can slope be zero? What does it signify?
Yes. A zero slope means there is no vertical change as x changes. Such a line is parallel to the x-axis and is called a horizontal line.
5. Is slope dependent on the choice of points?
No. For a straight line, the slope remains the same irrespective of which two distinct points on the line are chosen.
6. How is slope related to angle of inclination?
Slope is the tangent of the angle of inclination: [m = \tan \theta]. Larger the angle, steeper the line.
7. What is the slope of a line making 45° with x-axis?
[m = \tan 45° = 1]. The line rises one unit vertically for every one unit horizontally.
8. Can slope be a fraction or irrational number?
Yes. Slope can be rational, irrational, positive, negative, or zero, depending on the orientation of the line.
9. What does a very large slope indicate?
A very large slope indicates that the line is almost vertical, meaning a very small horizontal change causes a large vertical change.
10. Is slope a vector quantity?
No. Slope is a scalar quantity. It has magnitude and sign, but no direction in space like a vector.
11. Why do parallel lines have equal slopes?
Parallel lines have the same inclination with the x-axis, hence their angles [\theta] are equal, giving equal slopes [m = \tan \theta].
12. Why do perpendicular lines satisfy [m₁m₂ = −1]?
If two lines are perpendicular, the angle between them is 90°. Using trigonometric identities, this leads to the condition [m₁m₂ = −1].
13. What is the slope of x-axis?
The x-axis is a horizontal line, hence its slope is [0].
14. What is the slope of y-axis?
The y-axis is a vertical line, so its slope is not defined.
15. Can two different lines have the same slope?
Yes. Such lines are parallel and never intersect.
6. FAQ / Common Misconceptions
1. Is slope always positive?
No. Slope can be positive, negative, zero, or undefined.
2. Is slope the same as y-intercept?
No. Slope describes inclination, while y-intercept describes where the line cuts the y-axis.
3. Does slope depend on the length of the line?
No. Slope depends only on orientation, not on the length of the line segment.
4. Can slope be infinite?
No. Vertical lines have undefined slope, not infinite slope.
5. Can slope change along a straight line?
No. If slope changes, the graph is not a straight line.
6. Is slope affected by shifting the line?
No. Translating a line parallel to itself does not change its slope.
7. Can two intersecting lines have the same slope?
No. If slopes are equal, lines are parallel and do not intersect.
8. Is slope always equal to tanθ?
Yes, provided θ is the angle made with the positive x-axis.
9. Why is slope dimensionless?
Because it is a ratio of two lengths measured in the same unit.
10. Can slope be greater than 1?
Yes. A slope greater than 1 indicates a steep rising line.
7. Practice Questions with Step-by-Step Solutions
Question 1. Find the slope of the line joining (2, 3) and (6, 11).
Step-by-Step Solution:
Given points:
[ (x₁, y₁) = (2, 3), (x₂, y₂) = (6, 11) ]
Apply slope formula:
[m = \dfrac{y₂ − y₁}{x₂ − x₁}]
Substitute values:
[m = \dfrac{11 − 3}{6 − 2}]
Simplify:
[m = \dfrac{8}{4} = 2]
Conclusion:
The slope of the line is [2].
Question 2. Find the slope of a line making an angle of 30° with the x-axis.
Step-by-Step Solution:
Given angle: [\theta = 30°]
Use formula:
[m = \tan \theta]
Substitute value:
[m = \tan 30°]
Evaluate:
[m = \dfrac{1}{\sqrt{3}}]
Conclusion:
The slope is [\dfrac{1}{\sqrt{3}}].
Question 3. Find the slope of the line x = 5.
Step-by-Step Solution:
Equation represents a vertical line.
For a vertical line, [\Delta x = 0].
Slope formula becomes undefined.
Conclusion:
Slope of the line is not defined.
Question 4. Find the slope of the line y = 7.
Step-by-Step Solution:
Equation represents a horizontal line.
No change in y-coordinate.
Hence, slope is zero.
Conclusion:
Slope [m = 0].
Question 5. Find the slope of the line passing through (−1, 4) and (3, −2).
Step-by-Step Solution:
Points: [ (−1, 4), (3, −2) ]
Apply formula:
[m = \dfrac{−2 − 4}{3 − (−1)}]
Simplify:
[m = \dfrac{−6}{4} = −\dfrac{3}{2}]
Conclusion:
The slope is [−\dfrac{3}{2}].
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