Q.Find the values of k for which the line (k−3)x−(4−k2)y+k2−7k+6=0 is
Concept understanding — Slope Calculation
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.
You can swap which point is first — just be consistent. Using (4,8) as (x1,y1) and (1,2) as (x2,y2) gives m=1−42−8=−3−6=2, the same result.
Why Slope Matters
Slope is the foundation of linear relationships. It tells you the rate of change — how one quantity changes as another changes. In physics, slope of a distance-time graph gives speed. In economics, slope of a cost line gives marginal cost. In geometry, slope determines whether lines are parallel (same slope) or perpendicular (slopes multiply to −1).
Once you see slope as "rise over run", you've unlocked the language of change.
Slope Calculation is one of the very first ideas introduced in the NCERT Class 11 Mathematics chapter on Straight Lines, and it's what students mean when they search "slope of a line formula class 11 maths" or "coordinate geometry important questions". Being fluent with rise-over-run also pays off directly in JEE Main and CET questions on lines, parallelism, and perpendicularity.
For Ax+By+C=0 with A=k−3, B=−(4−k2), C=k2−7k+6:
(a) Parallel to x-axis (A=0, B=0): k−3=0⇒k=3 (valid, B=5=0).
(b) Parallel to y-axis (B=0, A=0): 4−k2=0⇒k=±2 (both valid).
(c) Through the origin (C=0): k2−7k+6=0⇒(k−1)(k−6)=0⇒k=1 or 6.
- k=3
- k=2 or k=−2
- k=1 or k=6
The line is parallel to the x-axis when k=3; parallel to the y-axis when k=2 or k=−2; and passes through the origin when k=1 or k=6.
The given line is
(k−3)x−(4−k2)y+(k2−7k+6)=0,
which has the form Ax+By+C=0 with A=k−3, B=−(4−k2), C=k2−7k+6.
(a) Parallel to the x-axis
A line parallel to the x-axis is horizontal, so its slope is 0. For Ax+By+C=0, the slope is −A/B, which is 0 exactly when the coefficient of x vanishes (and B=0, so the line doesn't degenerate):
A=k−3=0 ⇒ k=3.
Check: B=−(4−9)=5=0, so this is valid.
(b) Parallel to the y-axis
A line parallel to the y-axis is vertical — it has no y-term, so the coefficient of y must vanish (with A=0):
B=−(4−k2)=0 ⇒ k2=4 ⇒ k=2 or k=−2.
Check the coefficient of x in each case: for k=2, A=2−3=−1=0; for k=−2, A=−2−3=−5=0. Both values are valid.
(c) Passing through the origin
A line passes through the origin (0,0) exactly when substituting x=0,y=0 satisfies the equation — i.e. the constant term is zero:
k2−7k+6=0 ⇒ (k−1)(k−6)=0 ⇒ k=1 or k=6.
- k=3
- k=2 or k=−2
- k=1 or k=6
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 y-axis into the second quadrant), its angle from the positive x-axis is:
θ=90°+30°=120°
Slope =tan120°=tan(180°−60°)=−tan60°=−3.
✓Final answer(a) −3.
- 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.
The slope of a line is m=tanθ, where θ is the angle the line makes with the positive X-axis (measured anticlockwise). A line parallel to the X-axis makes an angle θ=0° with it, so m=tan0°=0.
✓Final answerThe slope of any line parallel to the X-axis is 0, which is option (a).
- 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.
The x-axis is the horizontal reference line, so the angle it makes with the (positive direction of the) x-axis is θ=0∘.
Slope =tan0∘=0.
✓Final answerThe correct option is (c) 0.
- 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:
y−3=21(x+4)⇒2(y−3)=x+4⇒2y−6=x+4.
Rearranging: x−2y+10=0.
✓Final answerThe line is x−2y+10=0.
- 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.
Since both points share the same y-coordinate, the line is horizontal, confirming slope 0.
✓Final answerThe correct option is (a) 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
So the slope is 43, not 34 as claimed.
✓Final answerFalse — the slope is 43.
- 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
Comparing with y=mx+c, the slope is m=−2.
✓Final answerThe slope of the line 6x+3y−5=0 is −2.
- 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):
m=−1−34−(−2)=−46=−23.
✓Final answerThe slope of the line through (3,−2) and (−1,4) is −23.
- 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.
By definition, if a line makes an angle θ with the positive direction of the x-axis, its slope is m=tanθ. Here θ=30∘, and tan30∘=31.
✓Final answerThe correct option is (c) 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.
A slope of 0 means the line is horizontal (parallel to the x-axis), which makes sense since both points share the same y-value.
✓Final answerThe correct option is (a) 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.
Step 2. Comparing with y=mx+c, the slope m=−4.
✓Final answerThe slope of the line 4x+y+5=0 is −4.
- 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.
Step 2. Its inclination with the x-axis is 0°, and slope =tan(0°)=0.
✓Final answerThe slope of a horizontal line is 0.
🎓Unlock everything free for 14 days
- ✓Full step-by-step solutions
- ✓Concept-first explanations
- ✓Methods, shortcuts & mistakes
- ✓PYQ mapping + timed mock tests
Full access for 14 days. No credit card required.