Equations of Lines Parallel to Axes
In coordinate geometry, certain lines have very simple equations. A line parallel to the x-axis has the equation y = b, and a line parallel to the y-axis has the equation x = a.
These lines are special cases of linear equations in two variables. Understanding their equations is essential before studying the general equation of a line.
In the NCERT Class 9 syllabus, this topic connects the chapters on Linear Equations in Two Variables and Coordinate Geometry.
What is Equations of Lines Parallel to Axes?
Definition: A line parallel to the x-axis passes through all points with the same y-coordinate. Its equation is y = b, where b is a constant.
Definition: A line parallel to the y-axis passes through all points with the same x-coordinate. Its equation is x = a, where a is a constant.
Key facts:
- The x-axis itself has the equation y = 0.
- The y-axis itself has the equation x = 0.
- Lines parallel to the x-axis are horizontal.
- Lines parallel to the y-axis are vertical.
- Every point on y = 3 has y-coordinate 3, regardless of x. Examples: (0, 3), (2, 3), (−5, 3).
- Every point on x = −4 has x-coordinate −4, regardless of y. Examples: (−4, 0), (−4, 7), (−4, −2).
Equations of Lines Parallel to Axes Formula
Key Formulas:
1. Line parallel to the x-axis:
y = b (where b is a constant)
Properties:
- Slope = 0 (horizontal line)
- Perpendicular distance from x-axis = |b| units
- Above x-axis if b > 0, below if b < 0, on x-axis if b = 0
2. Line parallel to the y-axis:
x = a (where a is a constant)
Properties:
- Slope is undefined (vertical line)
- Perpendicular distance from y-axis = |a| units
- Right of y-axis if a > 0, left if a < 0, on y-axis if a = 0
3. General form connection:
- y = b can be written as 0x + 1y = b (coefficient of x is 0).
- x = a can be written as 1x + 0y = a (coefficient of y is 0).
Derivation and Proof
Why y = b represents a horizontal line:
Step 1: Consider the general form of a linear equation: ax + by = c.
Step 2: If a = 0 (coefficient of x is zero):
- The equation becomes by = c, or y = c/b = constant.
- The value of x does not appear — it can take any value.
- This means for every value of x, y remains the same.
- All points lie at the same height on the Cartesian plane.
Step 3: Plotting points confirms a horizontal line:
- y = 3: Points (0, 3), (1, 3), (2, 3), (−1, 3), (−2, 3) are all at height 3.
- Connecting them gives a straight horizontal line.
Why x = a represents a vertical line:
Step 1: If b = 0 (coefficient of y is zero):
- The equation becomes ax = c, or x = c/a = constant.
- The value of y does not appear — it can take any value.
- All points share the same x-coordinate.
Step 2: Plotting points confirms a vertical line:
- x = −2: Points (−2, 0), (−2, 1), (−2, 3), (−2, −4) all have x = −2.
- Connecting them gives a straight vertical line.
Types and Properties
Types of Lines Parallel to Axes:
1. Lines parallel to x-axis above it (y = b, b > 0):
- y = 1, y = 2, y = 5.5
- These lines lie in the upper half of the Cartesian plane.
- They pass through Quadrants I and II.
2. Lines parallel to x-axis below it (y = b, b < 0):
- y = −1, y = −3, y = −4.5
- These lines lie in the lower half of the Cartesian plane.
- They pass through Quadrants III and IV.
3. The x-axis itself (y = 0):
- Passes through the origin and all points with y = 0.
- Separates the upper and lower halves.
4. Lines parallel to y-axis to the right (x = a, a > 0):
- x = 1, x = 4, x = 6.5
- These lie in the right half of the Cartesian plane.
- They pass through Quadrants I and IV.
5. Lines parallel to y-axis to the left (x = a, a < 0):
- x = −2, x = −5, x = −3.5
- These lie in the left half.
- They pass through Quadrants II and III.
6. The y-axis itself (x = 0):
- Passes through the origin and all points with x = 0.
- Separates the left and right halves.
7. Intersection properties:
- Two horizontal lines (y = b₁ and y = b₂, b₁ ≠ b₂) are parallel and never intersect.
- A horizontal line (y = b) and a vertical line (x = a) always intersect at exactly one point: (a, b).
- Two vertical lines (x = a₁ and x = a₂, a₁ ≠ a₂) are parallel and never intersect.
Solved Examples
Example 1: Example 1: Write the equation of a line parallel to x-axis through (3, 5)
Problem: Write the equation of the line parallel to the x-axis that passes through the point (3, 5).
Solution:
Given:
- Point: (3, 5)
- Line is parallel to the x-axis.
Method:
- A line parallel to the x-axis has equation y = b.
- Since the line passes through (3, 5), the y-coordinate is 5.
- Therefore, b = 5.
Answer: The equation is y = 5.
Example 2: Example 2: Write the equation of a line parallel to y-axis through (−4, 7)
Problem: Write the equation of the line parallel to the y-axis passing through (−4, 7).
Solution:
Given:
- Point: (−4, 7)
- Line is parallel to the y-axis.
Method:
- A line parallel to the y-axis has equation x = a.
- Since the line passes through (−4, 7), the x-coordinate is −4.
- Therefore, a = −4.
Answer: The equation is x = −4.
Example 3: Example 3: Identify the line from its equation
Problem: Describe the graph of: (i) y = −3 (ii) x = 5 (iii) y = 0 (iv) x = 0.
Solution:
- (i) y = −3: Horizontal line, 3 units below the x-axis, passing through Quadrants III and IV.
- (ii) x = 5: Vertical line, 5 units to the right of the y-axis, passing through Quadrants I and IV.
- (iii) y = 0: This is the x-axis itself.
- (iv) x = 0: This is the y-axis itself.
Answer: (i) Horizontal line at y = −3 (ii) Vertical line at x = 5 (iii) x-axis (iv) y-axis.
Example 4: Example 4: Find three points on y = 4
Problem: List three points that lie on the line y = 4 and three points that do not.
Solution:
Points on y = 4 (y-coordinate must be 4):
- (0, 4), (3, 4), (−6, 4)
Points NOT on y = 4 (y-coordinate is not 4):
- (4, 0), (1, 3), (−2, 5)
Answer: On the line: (0, 4), (3, 4), (−6, 4). Not on the line: (4, 0), (1, 3), (−2, 5).
Example 5: Example 5: Find the intersection point
Problem: Find the point of intersection of the lines x = 3 and y = −2.
Solution:
Given:
- x = 3 (vertical line)
- y = −2 (horizontal line)
Method:
- At the intersection, both conditions hold: x = 3 and y = −2.
- The intersection point is (3, −2).
Answer: The intersection point is (3, −2).
Example 6: Example 6: Express 2y − 6 = 0 as a line parallel to axes
Problem: Express the equation 2y − 6 = 0 in standard form. What type of line does it represent?
Solution:
- 2y − 6 = 0
- 2y = 6
- y = 3
Interpretation:
- This is a horizontal line parallel to the x-axis, at a distance of 3 units above it.
- It passes through (0, 3), (1, 3), (−5, 3), etc.
Answer: y = 3. It is a horizontal line parallel to the x-axis.
Example 7: Example 7: Express 3x + 12 = 0 as a line parallel to axes
Problem: Express 3x + 12 = 0 in standard form. What type of line does it represent?
Solution:
- 3x + 12 = 0
- 3x = −12
- x = −4
Interpretation:
- This is a vertical line parallel to the y-axis, at a distance of 4 units to the left of it.
- It passes through (−4, 0), (−4, 5), (−4, −3), etc.
Answer: x = −4. It is a vertical line parallel to the y-axis.
Example 8: Example 8: Rectangle formed by axes-parallel lines
Problem: Find the area of the rectangle formed by the lines x = 2, x = 7, y = 1, and y = 6.
Solution:
Finding the vertices:
- Vertex 1: Intersection of x = 2 and y = 1 → (2, 1)
- Vertex 2: Intersection of x = 7 and y = 1 → (7, 1)
- Vertex 3: Intersection of x = 7 and y = 6 → (7, 6)
- Vertex 4: Intersection of x = 2 and y = 6 → (2, 6)
Dimensions:
- Length = 7 − 2 = 5 units
- Width = 6 − 1 = 5 units
Area = 5 × 5 = 25 sq units
Answer: The area of the rectangle is 25 square units.
Example 9: Example 9: Distance between parallel lines
Problem: Find the distance between the lines y = 3 and y = −5.
Solution:
Given:
- Both are horizontal lines (parallel to x-axis and to each other).
Distance:
- Distance = |3 − (−5)| = |3 + 5| = 8 units
Answer: The distance is 8 units.
Example 10: Example 10: Line through the origin parallel to y = 4
Problem: A line passes through the origin and is parallel to y = 4. Write its equation.
Solution:
Analysis:
- y = 4 is a horizontal line.
- A line parallel to it is also horizontal, with equation y = b.
- Since it passes through the origin (0, 0), the y-coordinate is 0.
- Therefore, b = 0.
Answer: The equation is y = 0 (the x-axis).
Real-World Applications
Applications of Lines Parallel to Axes:
- Graphing linear equations: The simplest linear equations to graph are y = b and x = a. They serve as starting points for understanding more complex equations.
- Coordinate geometry: Forming rectangles and squares in the Cartesian plane by using pairs of horizontal and vertical lines.
- Maps and grids: Latitude lines (horizontal) correspond to y = b, and longitude lines (vertical) correspond to x = a.
- Computer graphics: Defining clipping boundaries, grid lines, and alignment guides using horizontal and vertical lines.
- Construction and architecture: Level lines (horizontal) and plumb lines (vertical) are essential for building alignment.
- Graph interpretation: Horizontal lines represent constant values (no change). In physics, a horizontal line on a velocity-time graph means constant velocity.
Key Points to Remember
- A line parallel to the x-axis has equation y = b. It is horizontal with slope 0.
- A line parallel to the y-axis has equation x = a. It is vertical with undefined slope.
- The x-axis itself is y = 0. The y-axis itself is x = 0.
- In y = b, the value of x can be any real number.
- In x = a, the value of y can be any real number.
- A horizontal line y = b and a vertical line x = a always intersect at (a, b).
- Two horizontal lines are parallel if they have different values of b.
- Two vertical lines are parallel if they have different values of a.
- The distance between parallel horizontal lines y = b₁ and y = b₂ is |b₁ − b₂|.
- The distance between parallel vertical lines x = a₁ and x = a₂ is |a₁ − a₂|.
Practice Problems
- Write the equation of the line parallel to the x-axis passing through (−7, 3).
- Write the equation of the line parallel to the y-axis passing through (5, −1).
- The line 4y − 20 = 0 is parallel to which axis? Find three points on it.
- Find the point of intersection of x = −6 and y = 8.
- Find the area of the rectangle formed by the lines x = −3, x = 4, y = 2, y = 9.
- A horizontal line passes through (0, −5). Write its equation and find the distance from the x-axis.
- Express 5x + 15 = 0 as a line parallel to one of the axes. Which quadrants does it pass through?
- Write the equations of all four sides of a square with vertices at (1, 1), (1, 6), (6, 6), (6, 1).
Frequently Asked Questions
Q1. What is the equation of a line parallel to the x-axis?
The equation is y = b, where b is a constant. All points on this line have the same y-coordinate equal to b.
Q2. What is the equation of a line parallel to the y-axis?
The equation is x = a, where a is a constant. All points on this line have the same x-coordinate equal to a.
Q3. What is the slope of y = 5?
The slope is 0. All horizontal lines (parallel to the x-axis) have slope 0 because there is no vertical change as you move along the line.
Q4. What is the slope of x = 3?
The slope is undefined. Vertical lines have undefined slope because the horizontal change (run) is zero, and division by zero is undefined.
Q5. Is y = 0 the equation of the x-axis or y-axis?
y = 0 is the equation of the x-axis. Every point on the x-axis has y-coordinate equal to 0. The y-axis has equation x = 0.
Q6. Can a line parallel to the x-axis pass through the origin?
Yes. The line y = 0 (the x-axis itself) is parallel to the x-axis and passes through the origin (0, 0).
Q7. Do the lines x = 3 and x = −3 intersect?
No. Both are vertical lines (parallel to the y-axis) at different positions. Parallel lines never intersect.
Q8. How do you find the equation of a horizontal line through a given point?
Take the y-coordinate of the given point. The equation is y = (that y-coordinate). For example, through (4, −7), the horizontal line is y = −7.
Q9. Is this topic in CBSE Class 9?
Yes. Equations of lines parallel to axes are covered in Chapter 4 (Linear Equations in Two Variables) of the CBSE Class 9 Mathematics syllabus.
Q10. What is the general form of a line parallel to an axis?
Parallel to x-axis: 0x + 1y = b (coefficient of x is 0). Parallel to y-axis: 1x + 0y = a (coefficient of y is 0).
