Q.The angle between the x-axis & the line joining the points (3,−1) and (4,−2) is
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Slope Calculation — From Intuition to Precision
Imagine you're walking up a hill. Some hills are gentle — you barely notice the climb. Others are so steep you have to lean forward and use your hands. That "steepness" is what slope measures. In mathematics, slope tells us how fast a line rises or falls as we move from left to right.
The Intuition: Rise Over Run
Take any two points on a straight line. As you walk from the left point to the right point, two things happen:
- You move horizontally — that's the run.
- You move vertically — that's the rise (upwards) or fall (downwards).
Slope is simply the ratio:
Slope = (vertical change) ÷ (horizontal change)
If you climb 3 metres while walking 5 metres forward, the slope is 3/5=0.6. If you descend 2 metres while walking 4 metres forward, the slope is −2/4=−0.5 — negative because you're going downhill.
The Precise Definition
Given two distinct points (x1,y1) and (x2,y2) on a non-vertical line, the slope m is:
m=x2−x1y2−y1
The numerator is the rise (change in y), the denominator is the run (change in x). The order matters: subtract the first point's coordinates from the second's, consistently.
Never divide by zero. If x2=x1, the line is vertical — slope is undefined (not zero, not infinite — just undefined).
What the Number Tells You
| Slope value | What the line does |
|---|---|
| m>0 | Rises left to right (uphill) |
| m<0 | Falls left to right (downhill) |
| m=0 | Horizontal (flat) |
| m undefined | Vertical (straight up/down) |
The larger the absolute value ∣m∣, the steeper the line. A slope of 5 is much steeper than a slope of 0.2.
A Worked Example
Find the slope of the line through (1,2) and (4,8).
Step 1: Label the points. Let (x1,y1)=(1,2) and (x2,y2)=(4,8).
Step 2: Compute the rise: y2−y1=8−2=6.
Step 3: Compute the run: x2−x1=4−1=3.
Step 4: Divide: m=36=2.
The line rises 2 units vertically for every 1 unit it moves right. …
Slope of the line through (3,−1) and (4,−2) is 4−3−2−(−1)=−1, whose magnitude gives an inclination of tan−1(1)=45∘ with the …
Compute the slope of the line through the two points, then find the angle whose tangent equals the magnitude of that slope.
The slope of the line joining (x1,y1)=(3,−1) and (x2,y2)=(4,−2) is
m=x2−x1y2−y1=4−3−2−(−1)=1−1=−1
…
Showing the 12 most recent of 21 on this concept.
- CBSE 2026Set ANNUAL1 markMCQQ.Slope of the line which makes an angle of 30° with the positive direction of y-axis measured anticlockwise is(a) −3(b) −31(c) 31(d) 3
›Reveal solutionSolution
Convert the angle measured from the y-axis to the equivalent angle measured (anticlockwise) from the x-axis, then take its tangent as the slope.
Slope is defined as tanθ where θ is the angle the line makes with the positive x-axis, measured anticlockwise. The positive y-axis itself is at 90° from the positive x-axis. If the line makes a further 30° with the positive y-axis (measured anticlockwise, i.e. swinging past the …
- CBSE 2026Set ANNUAL1 markMCQQ.The slope of any line parallel to X-axis is:(a) 0(b) 1(c) −1(d) Not defined.
›Reveal solutionSolution
A line parallel to the X-axis is horizontal, so its slope m=tanθ with θ=0°, giving m=0.
…
- CBSE 2026Set ANNUAL1 markMCQQ.The slope of x-axis is:(a) 90∘(b) 1(c) 0(d) 45∘
›Reveal solutionSolution
Slope of a line =tanθ, where θ is the angle it makes with the positive x-axis; the x-axis itself makes a 0∘ angle with itself.
…
- CBSE 2026Set 1A1 markQ.Find the equation of the line which is passing through the point (−4,3) with slope 21.
›Reveal solutionSolution
Point-slope form through (−4,3) with slope 21 gives x−2y+10=0.
Using y−y1=m(x−x1) with (x1,y1)=(−4,3) and m=21: …
- CBSE 2025Set ANNUAL1 markMCQQ.The slope of the line passing through the points (3,−2) and (7,−2) is(a) 0(b) 1(c) 4(d) -1
›Reveal solutionSolution
The slope is 0 — the line through these points is horizontal.
The slope formula is m=x2−x1y2−y1.
For (3,−2) and (7,−2): m=7−3−2−(−2)=40=0.
…
- CBSE 2025Set ANNUAL1 markQ.The slope of the line 3x − 4y + 10 = 0 is 4/3. (True/False)
›Reveal solutionSolution
Writing the line in slope-intercept form gives the true slope, which does not match the statement.
Line: 3x−4y+10=0
Solve for y: 4y=3x+10⇒y=43x+410
…
- CBSE 2025Set ANNUAL1 markQ.Fill in the blank: The slope of the line 6x+3y−5=0 will be ____.
›Reveal solutionSolution
Rearranging 6x+3y−5=0 into y=mx+c form shows the slope m=−2.
Starting from 6x+3y−5=0, solve for y:
3y=−6x+5
y=−2x+35
…
- CBSE 2025Set ANNUAL1 markQ.Find the slope of the line passing through the points (3,−2) and (−1,4).
›Reveal solutionSolution
Using the two-point slope formula on (3,−2) and (−1,4) gives slope −23.
Slope m=x2−x1y2−y1.
With (x1,y1)=(3,−2) and (x2,y2)=(−1,4):
…
- CBSE 2024Set ANNUAL1 markMCQQ.A line makes an angle of 30∘ with the positive direction of x-axis. Find the slope of the line —(a) 23(b) 3(c) 31(d) 32
›Reveal solutionSolution
The slope is tan30∘=31.
…
- CBSE 2024Set ANNUAL1 markMCQQ.The slope of a line passing through the points (3,−2) and (7,−2) is:(a) 0(b) 1(c) −1(d) Not defined.
›Reveal solutionSolution
The slope of the line through (3,−2) and (7,−2) is 0.
Slope formula: m=x2−x1y2−y1. With (x1,y1)=(3,−2) and (x2,y2)=(7,−2):
m=7−3−2−(−2)=40=0. …
- CBSE 2024Set ANNUAL1 markQ.Fill in the blank: The slope of the line 4x+y+5=0 is ______.
›Reveal solutionSolution
Rearranging 4x+y+5=0 into y=mx+c form shows the slope is −4.
Step 1. Rearrange: y=−4x−5.
…
- CBSE 2024Set ANNUAL1 markQ.Write the slope of a horizontal line.
›Reveal solutionSolution
A horizontal line has y constant, so its slope (rate of change of y with x) is 0.
Step 1. A horizontal line has the form y=c for a constant c.
…
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