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CBSE · Class 11 · Mathematics

Equations of a Line (Ch 9: 9.3)

4 concepts · 32 practice questions with answers · Free to read

Horizontal, vertical and point-slope form

From daily life

Standing on a hill at a known spot, and knowing how steep the path is, you can describe the whole path. That is point-slope form.

Simple definition

A horizontal line through (0, b) has equation y = b; a vertical line through (a, 0) has equation x = a. The line through (x₁, y₁) with slope m is y − y₁ = m(x − x₁).

Formula: y − y₁ = m (x − x₁)

-10123456(2, 3)

In the figure: The line through (2, 3) with slope 3/2.

Rule 1: Use y − y₁ = m(x − x₁) — Substitute the point and slope, then simplify.

  • Easy: Through (2, 3), slope 4: y − 3 = 4(x − 2), i.e. y = 4x − 5.
  • Medium: Through (−1, 5), slope −2: y − 5 = −2(x + 1), i.e. y = −2x + 3.
  • Exam-type: Through (4, 0), slope 1/2: y = (x − 4)/2.

Rule 2: Horizontal and vertical lines — Only one coordinate is fixed.

  • Easy: Through (3, −2), parallel to the x-axis: y = −2.
  • Medium: Through (3, −2), parallel to the y-axis: x = 3.
  • Exam-type: The x-axis is y = 0; the y-axis is x = 0.
Remember

Parallel to x-axis: y = constant. Parallel to y-axis: x = constant.

Common mistake

Writing y + y₁ instead of y − y₁ when the point has negative coordinates: be careful with signs.

Memory clue

Point and slope: y minus y-one equals m times x minus x-one.

Practice questions

Easy
  1. 1Write the equation of the line through (1, 2) with slope 3.
  2. 2Write the line through (0, 5) with slope −1.
  3. 3Write the line through (3, 4) parallel to the x-axis.
Medium
  1. 4Write the line through (3, 4) parallel to the y-axis.
  2. 5Write the line through (−2, 1) with slope 1/2.
  3. 6Write the line through (2, 3) that makes 45° with the positive x-axis.
Challenging
  1. 7Write the line through (1, 1) parallel to the line y = 2x + 5.
  2. 8Write the line through (2, 3) perpendicular to y = 2x.
Show answers (try them yourself first)
  1. 1y − 2 = 3(x − 1), i.e. y = 3x − 1
  2. 2y = −x + 5
  3. 3y = 4
  4. 4x = 3
  5. 5y − 1 = (x + 2)/2, i.e. 2y = x + 4
  6. 6m = 1: y − 3 = x − 2, i.e. y = x + 1
  7. 7y − 1 = 2(x − 1), i.e. y = 2x − 1
  8. 8slope −1/2: y − 3 = −(x − 2)/2, i.e. x + 2y = 8

Two-point form

From daily life

Two pegs in a wall fix a straight string. Two points always determine exactly one line.

Simple definition

The line through (x₁, y₁) and (x₂, y₂) is y − y₁ = ((y₂ − y₁)/(x₂ − x₁)) (x − x₁).

Formula: y − y₁ = [(y₂ − y₁)/(x₂ − x₁)] (x − x₁)

Rule 1: Line through two points — Find the slope, then use point-slope form.

  • Easy: (1, 2) and (3, 8): m = 3, y − 2 = 3(x − 1), y = 3x − 1.
  • Medium: (2, 5) and (4, 1): m = −2, y − 5 = −2(x − 2), y = −2x + 9.
  • Exam-type: (0, 0) and (6, 4): m = 2/3, y = 2x/3.

Rule 2: Check and use — Substitute the second point to check.

  • Easy: y = 3x − 1: at (3, 8): 3(3) − 1 = 8, correct.
  • Medium: Does (5, 7) lie on the line through (1, 3) and (3, 5)? y = x + 2 gives 7: yes.
  • Exam-type: Find k if (4, k) lies on y = 2x − 3: k = 5.
Remember

Use any one of the two points; the result is the same.

Common mistake

Mixing the order of subtraction in numerator and denominator.

Memory clue

Slope first, then point-slope.

Practice questions

Easy
  1. 1Find the line through (0, 1) and (2, 5).
  2. 2Find the line through (1, 4) and (3, 2).
  3. 3Find the line through (−1, 0) and (1, 4).
Medium
  1. 4Find the line through (2, 3) and (2, 7).
  2. 5Find the line through (0, 0) and (5, 10).
  3. 6The line through (1, 3) and (3, 7) also passes through (2, k). Find k.
Challenging
  1. 7Find the line joining the midpoint of (0, 0), (4, 6) to (1, 1).
  2. 8The vertices of a triangle are A(0, 0), B(4, 0), C(2, 6). Find the equation of AC.
Show answers (try them yourself first)
  1. 1y = 2x + 1
  2. 2y = −x + 5
  3. 3y = 2x + 2
  4. 4vertical line x = 2
  5. 5y = 2x
  6. 6Slope = 2, line y = 2x + 1, so k = 5
  7. 7midpoint (2, 3); slope 2; y = 2x − 1
  8. 8y = 3x

Slope-intercept and intercept form

From daily life

A taxi fare: ₹10 flat plus ₹12 per km. The graph is a line y = 12x + 10; 12 is the slope and 10 is where it cuts the y-axis.

Simple definition

The line with slope m and y-intercept c is y = mx + c. A line making intercepts a on the x-axis and b on the y-axis is x/a + y/b = 1.

Formula: y = mx + c; x/a + y/b = 1

-10123456(a, 0)(0, b)

In the figure: The line cuts the x-axis at a = 3 and the y-axis at b = 2: x/3 + y/2 = 1.

Rule 1: Slope-intercept form — Read m and c.

  • Easy: y = 3x + 2: slope 3, y-intercept 2.
  • Medium: Slope −1, y-intercept 4: y = −x + 4.
  • Exam-type: y = 5: slope 0, y-intercept 5.

Rule 2: Intercept form — Divide by the constant to get 1 on the right.

  • Easy: x-intercept 4, y-intercept 3: x/4 + y/3 = 1, i.e. 3x + 4y = 12.
  • Medium: 3x + 2y = 6 → x/2 + y/3 = 1: intercepts 2 and 3.
  • Exam-type: x-intercept −2, y-intercept 5: x/(−2) + y/5 = 1.
Remember

Put x = 0 to find the y-intercept and y = 0 to find the x-intercept.

Common mistake

Forgetting that the intercept form needs the right-hand side to be 1.

Memory clue

x over a plus y over b equals one.

Practice questions

Easy
  1. 1Find the slope and y-intercept of y = −2x + 7.
  2. 2Write the line with slope 4 and y-intercept −3.
  3. 3Write the line with x-intercept 5 and y-intercept 2.
Medium
  1. 4Find the intercepts of 4x + 3y = 24.
  2. 5A line cuts off intercepts 3 and −3. Find its equation and slope.
  3. 6Find the line through (2, 3) with equal intercepts on both axes.
Challenging
  1. 7The line through (1, 2) cuts the axes at A and B so that (1, 2) is the midpoint of AB. Find the line.
  2. 8Find the area of the triangle formed by x/4 + y/3 = 1 and the axes.
Show answers (try them yourself first)
  1. 1−2 and 7
  2. 2y = 4x − 3
  3. 3x/5 + y/2 = 1, i.e. 2x + 5y = 10
  4. 4x-intercept 6, y-intercept 8
  5. 5x/3 − y/3 = 1, i.e. y = x − 3; slope 1
  6. 6x/a + y/a = 1 and 2 + 3 = a, so x + y = 5
  7. 7A(2, 0), B(0, 4): x/2 + y/4 = 1, i.e. 2x + y = 4
  8. 8(1/2)(4)(3) = 6 square units

General equation of a line

From daily life

Different people write the same road as 2x + 3y = 6, y = −2x/3 + 2 or x/3 + y/2 = 1. The general form Ax + By + C = 0 covers every case.

Simple definition

Every line has an equation Ax + By + C = 0 with A and B not both 0. For B ≠ 0: slope = −A/B, y-intercept = −C/B; for A ≠ 0: x-intercept = −C/A.

Formula: Ax + By + C = 0

Rule 1: Convert — Solve for y to get slope-intercept form.

  • Easy: 2x + 3y − 6 = 0: y = −2x/3 + 2, slope −2/3.
  • Medium: 3x − y + 4 = 0: y = 3x + 4, slope 3.
  • Exam-type: x − 2y = 0: y = x/2, slope 1/2, passes through the origin.

Rule 2: Parallel and perpendicular lines — Compare slopes −A/B.

  • Easy: 2x + 3y = 5 and 4x + 6y = 1: both slope −2/3, parallel.
  • Medium: x + 2y = 3 and 2x − y = 1: slopes −1/2 and 2, product −1, perpendicular.
  • Exam-type: Line through (1, 1) parallel to x + 2y = 5: x + 2y = 3.
Remember

Lines parallel to Ax + By + C = 0 have the form Ax + By + k = 0.

Common mistake

Taking slope = A/B instead of −A/B.

Memory clue

Slope is minus A over B.

Practice questions

Easy
  1. 1Find the slope of 3x + 4y = 12.
  2. 2Write 2x − y + 5 = 0 in the form y = mx + c.
  3. 3Find the intercepts of 5x + 2y = 10.
Medium
  1. 4Are 3x + y = 4 and 6x + 2y = 9 parallel?
  2. 5Find the line through (2, 1) parallel to 3x + 4y = 7.
  3. 6Find the line through (2, 1) perpendicular to 3x + 4y = 7.
Challenging
  1. 7For which k is kx + 2y = 5 perpendicular to 2x − 3y = 1?
  2. 8Find the value of k if (3, 2) lies on 2x − ky + 4 = 0.
Show answers (try them yourself first)
  1. 1−3/4
  2. 2y = 2x + 5
  3. 3x-intercept 2, y-intercept 5
  4. 4Both have slope −3: yes
  5. 53x + 4y = 10
  6. 64x − 3y = 5
  7. 7slopes −k/2 and 2/3; product −1 gives k = 3
  8. 86 − 2k + 4 = 0, so k = 5