Slope or Gradient of a Straight Line
Finding the Slope of a Straight Line
🔹 If a straight line makes an angle θ with the positive direction of the x-axis, measured anticlockwise, then the value of tanθ is called the slope or gradient of the line.
Slope = m = tanθ
Special Cases:
⭐ The inclination θ is measured anticlockwise from the positive x-axis and is generally taken from 0° to less than 180°.
⭐ If a straight line makes an acute angle with the positive x-axis, its slope is positive.
⭐ If a straight line makes an obtuse angle with the positive x-axis, its slope is negative.
Lines Parallel to the Coordinate Axes
⭐ If a straight line is parallel to the x-axis, its slope is zero. Thus, if θ = 0°, then m = 0 (∵ tan 0° = 0).
⭐ If a straight line is parallel to the y-axis, its slope is undefined. Here θ = 90°, and since tan 90° is undefined, the slope has no finite value.
Slope of a Line Making Equal Angles with the Axes
⭐ If θ = 45°, then m = 1 (∵ tan 45° = 1).
⭐ If θ = 135°, then m = -1 (∵ tan 135° = -1).
Slope of the Line Joining Two Points
🔹 If a straight line joins the points P(x1, y1) and Q(x2, y2), then its slope m or tanθ is
∴
Angle Between Two Straight Lines
∴
🔹 If the slopes of two straight lines are m1 and m2, the smaller angle between them is θ.
Special Cases:
🔹 The two lines are parallel when m1 = m2.
🔹 The two lines are perpendicular when m1m2 = -1, provided both slopes are finite.
🧠 Quick Formula Revision
| Topic | Formula / Result |
|---|---|
| Slope from inclination | m = tanθ |
| Slope from two points | |
Line parallel to the x-axis | m = 0 |
Line parallel to the y-axis | Slope is undefined |
| Parallel lines | m₁ = m₂ |
| Perpendicular lines | m₁m₂ = −1 |
| Angle between two lines |
📚 Important Points
🔹 The inclination θ is generally taken in the range 0° ≤ θ < 180°.
🔹 If 0° < θ < 90°, the slope is positive. If 90° < θ < 180°, the slope is negative.
🔹 When θ = 90°, it is not mathematically correct to say that the slope is infinite. The slope is undefined because tan90° is undefined.
🔹 In the two-point formula, if x₂ − x₁ = 0, the line is vertical and its slope is undefined.
🔹 Use the condition m₁m₂ = −1 directly only when both lines have finite slopes.
🔹 To find the smaller angle between two lines, it is convenient to use the absolute value in the tangent formula.
✏️ Solved Examples
Example 1: Slope from Two Points
Find the slope of the line joining A(2, 3) and B(6, 11).
Therefore, the slope is m = 2.
Example 2: Checking Whether Two Lines Are Parallel
The slopes of two lines are 3 and 3. Since m₁ = m₂, the two lines are parallel.
Example 3: Slope of a Perpendicular Line
If the slope of a line is 2, find the slope of a line perpendicular to it.
m₁m₂ = −1 2m₂ = −1 ⇒ m₂ = −1/2Therefore, the slope of the perpendicular line is −1/2.
