Abscissa and Ordinate
In coordinate geometry, every point on the Cartesian plane is identified by an ordered pair (x, y). The two numbers in this pair have specific names — the abscissa and the ordinate.
The abscissa is the x-coordinate of a point. It measures the perpendicular distance of the point from the y-axis. The ordinate is the y-coordinate. It measures the perpendicular distance from the x-axis.
Understanding abscissa and ordinate is fundamental to reading, plotting, and interpreting points on the Cartesian plane. These terms form the basis of all coordinate geometry studied in Class 9 and beyond.
The Cartesian coordinate system was introduced by René Descartes in the 17th century. It unified algebra and geometry by allowing geometric shapes to be described using algebraic equations. Every point, line, and curve can be represented using coordinates.
In the NCERT Class 9 textbook, students learn to identify coordinates, plot points in all four quadrants, and understand the significance of the signs of the abscissa and ordinate in determining the quadrant in which a point lies.
What is Abscissa and Ordinate?
Definition: For any point P(x, y) on the Cartesian plane:
- The abscissa is the first element x of the ordered pair. It gives the horizontal distance from the y-axis.
- The ordinate is the second element y of the ordered pair. It gives the vertical distance from the x-axis.
Point P(x, y) → Abscissa = x, Ordinate = y
Special Cases:
- For any point on the x-axis, the ordinate is 0. The point has the form (x, 0).
- For any point on the y-axis, the abscissa is 0. The point has the form (0, y).
- The origin O has both abscissa and ordinate equal to 0: O(0, 0).
Important:
- The abscissa is positive to the right of the y-axis and negative to the left.
- The ordinate is positive above the x-axis and negative below it.
- The order matters: (3, 5) and (5, 3) are different points because the abscissa and ordinate are swapped.
Abscissa and Ordinate Formula
Key Facts and Notation:
1. Ordered Pair Notation:
P(x, y) = P(abscissa, ordinate)
2. Sign Convention by Quadrant:
- Quadrant I: abscissa (+), ordinate (+) → P(+, +)
- Quadrant II: abscissa (−), ordinate (+) → P(−, +)
- Quadrant III: abscissa (−), ordinate (−) → P(−, −)
- Quadrant IV: abscissa (+), ordinate (−) → P(+, −)
3. Points on Axes:
- On x-axis: ordinate = 0, so the point is (x, 0)
- On y-axis: abscissa = 0, so the point is (0, y)
- Origin: abscissa = 0 and ordinate = 0, so the point is (0, 0)
4. Distance from Axes:
- Distance of P(x, y) from the y-axis = |x| = |abscissa|
- Distance of P(x, y) from the x-axis = |y| = |ordinate|
Derivation and Proof
How to Determine Abscissa and Ordinate from a Graph:
Step-by-step method to read coordinates of a plotted point:
- From the given point, draw a perpendicular to the x-axis. The foot of this perpendicular on the x-axis gives the abscissa.
- From the same point, draw a perpendicular to the y-axis. The foot of this perpendicular on the y-axis gives the ordinate.
- Write the coordinates as an ordered pair (abscissa, ordinate).
How to Plot a Point Given Its Abscissa and Ordinate:
- Locate the abscissa value on the x-axis.
- From this point, move vertically (up if ordinate is positive, down if negative) by the ordinate value.
- Mark the point where you stop. This is the required point P(x, y).
Why the Order Matters:
- The pair (3, 7) means: abscissa = 3 (move 3 units right), ordinate = 7 (move 7 units up).
- The pair (7, 3) means: abscissa = 7 (move 7 units right), ordinate = 3 (move 3 units up).
- These are two different points on the plane. The pair is ordered, not interchangeable.
Types and Properties
Types of Points Based on Abscissa and Ordinate:
1. Points in the Four Quadrants
- Quadrant I (e.g., (4, 5)): Both abscissa and ordinate are positive. The point lies to the right and above the origin.
- Quadrant II (e.g., (−3, 6)): Abscissa is negative, ordinate is positive. The point lies to the left and above the origin.
- Quadrant III (e.g., (−2, −4)): Both are negative. The point lies to the left and below the origin.
- Quadrant IV (e.g., (5, −3)): Abscissa is positive, ordinate is negative. The point lies to the right and below the origin.
2. Points on the X-axis
- The ordinate is always 0.
- Examples: (3, 0), (−5, 0), (0, 0).
- All such points lie along the horizontal axis.
3. Points on the Y-axis
- The abscissa is always 0.
- Examples: (0, 4), (0, −7), (0, 0).
- All such points lie along the vertical axis.
4. The Origin
- Both abscissa and ordinate are 0.
- It is the intersection of the x-axis and y-axis.
- Coordinates: (0, 0).
5. Mirror Points
- Reflection in x-axis: P(a, b) → P′(a, −b). The abscissa stays the same; the ordinate changes sign.
- Reflection in y-axis: P(a, b) → P′(−a, b). The ordinate stays the same; the abscissa changes sign.
- Reflection in the origin: P(a, b) → P′(−a, −b). Both change sign.
Solved Examples
Example 1: Example 1: Identify abscissa and ordinate
Problem: Write the abscissa and ordinate of the point P(7, −3).
Solution:
- The given point is P(7, −3).
- Abscissa = first coordinate = 7
- Ordinate = second coordinate = −3
Answer: Abscissa = 7, Ordinate = −3
Example 2: Example 2: Determine the quadrant
Problem: In which quadrant does the point A(−4, 6) lie? State its abscissa and ordinate.
Solution:
- Abscissa = −4 (negative) → left of y-axis
- Ordinate = 6 (positive) → above x-axis
- Negative abscissa + positive ordinate → Quadrant II
Answer: A(−4, 6) lies in Quadrant II. Abscissa = −4, Ordinate = 6.
Example 3: Example 3: Points on the axes
Problem: Where do the following points lie: B(0, −5), C(3, 0), D(0, 0)?
Solution:
- B(0, −5): Abscissa = 0 → point lies on the y-axis, 5 units below the origin.
- C(3, 0): Ordinate = 0 → point lies on the x-axis, 3 units to the right of the origin.
- D(0, 0): Both are 0 → point is the origin.
Answer: B is on the y-axis, C is on the x-axis, D is the origin.
Example 4: Example 4: Find the point from description
Problem: A point has abscissa −6 and ordinate 8. Write its coordinates and state the quadrant.
Solution:
- Abscissa = −6, Ordinate = 8
- Coordinates = (−6, 8)
- Abscissa is negative, ordinate is positive → Quadrant II
Answer: The point is (−6, 8) and it lies in Quadrant II.
Example 5: Example 5: Distance from axes
Problem: Find the distance of the point P(−4, 3) from the x-axis and the y-axis.
Solution:
- Distance from x-axis = |ordinate| = |3| = 3 units
- Distance from y-axis = |abscissa| = |−4| = 4 units
Answer: Distance from x-axis = 3 units; Distance from y-axis = 4 units.
Example 6: Example 6: Points with equal abscissa
Problem: Plot three points whose abscissa is 4. What do you observe?
Solution:
Three points with abscissa 4:
- A(4, 1), B(4, −3), C(4, 5)
Observation:
- All three points have the same x-coordinate (abscissa = 4).
- When plotted, they all lie on the vertical line x = 4.
- Points with the same abscissa are collinear and lie on a vertical line parallel to the y-axis.
Answer: The points A(4, 1), B(4, −3), and C(4, 5) all lie on the vertical line x = 4.
Example 7: Example 7: Reflection and coordinates
Problem: The reflection of point P(5, −2) in the x-axis is P′. Find the coordinates of P′ and state the abscissa and ordinate of P′.
Solution:
- When reflecting in the x-axis, the abscissa stays the same and the ordinate changes sign.
- P(5, −2) → P′(5, 2)
- Abscissa of P′ = 5
- Ordinate of P′ = 2
Answer: P′ = (5, 2). Abscissa = 5, Ordinate = 2.
Example 8: Example 8: Finding a point given conditions
Problem: A point lies in Quadrant III. Its abscissa is −7 and its distance from the x-axis is 4. Find the point.
Solution:
- Abscissa = −7
- Distance from x-axis = |ordinate| = 4 → ordinate = ±4
- Since the point is in Quadrant III, both coordinates are negative.
- Therefore ordinate = −4
Answer: The point is (−7, −4).
Example 9: Example 9: Identify coordinates from a graph description
Problem: A point Q is 6 units to the left of the y-axis and 2 units above the x-axis. Find its abscissa and ordinate.
Solution:
- 6 units to the left of the y-axis → abscissa = −6
- 2 units above the x-axis → ordinate = 2
Answer: Q = (−6, 2). Abscissa = −6, Ordinate = 2.
Example 10: Example 10: Compare two ordered pairs
Problem: Are the points (4, 9) and (9, 4) the same? Justify using abscissa and ordinate.
Solution:
- Point A(4, 9): abscissa = 4, ordinate = 9
- Point B(9, 4): abscissa = 9, ordinate = 4
- The abscissa of A ≠ abscissa of B (4 ≠ 9)
- The ordinate of A ≠ ordinate of B (9 ≠ 4)
Two ordered pairs are equal only if their corresponding components are equal. Since both components differ, these are different points.
Answer: No, (4, 9) and (9, 4) are not the same point. The ordered pair is not interchangeable.
Real-World Applications
Applications of Abscissa and Ordinate:
- Map Reading and GPS Navigation: Latitude and longitude work similarly to ordinate and abscissa. GPS devices identify locations using two coordinates, just as points are identified on the Cartesian plane. For example, a GPS coordinate like (28.6139°N, 77.2090°E) for Delhi gives an ordinate-like value (latitude) and an abscissa-like value (longitude).
- Data Visualisation: In bar graphs, line graphs, and scatter plots, the horizontal axis represents one variable (abscissa) and the vertical axis represents another (ordinate). Reading any data point requires identifying both coordinates. For example, in a temperature-time graph, the abscissa is the time (in hours) and the ordinate is the temperature (°C).
- Computer Graphics and Game Design: Every pixel on a screen has an (x, y) position. The abscissa gives the horizontal position and the ordinate gives the vertical position. Moving a character 5 units right increases its abscissa by 5; moving it 3 units up increases its ordinate by 3.
- Spreadsheet Addressing: Cells in a spreadsheet are identified by column (like abscissa) and row (like ordinate). For example, cell B3 corresponds to column 2, row 3 — a coordinate-like system.
- Physics — Motion Graphs: In distance-time or velocity-time graphs, the abscissa typically represents time and the ordinate represents distance or velocity. The slope of the curve at any point gives the speed or acceleration.
- Architecture and Engineering: Building plans use coordinate systems where every structural element (columns, walls, doorways) is placed at a specific (x, y) position relative to a reference corner of the building.
- Robotics and CNC Machines: Computer-controlled machines and robots navigate using coordinate systems. The controller sends abscissa and ordinate values to position the tool or arm at the exact required location.
- Medical Imaging: CT scans and MRI machines create cross-sectional images where each pixel is identified by its abscissa and ordinate within the scan plane. This allows precise localisation of tumours, fractures, and other features.
Key Points to Remember
- The abscissa is the x-coordinate (first value) of an ordered pair.
- The ordinate is the y-coordinate (second value) of an ordered pair.
- For point P(x, y): abscissa = x, ordinate = y.
- On the x-axis, the ordinate is always 0.
- On the y-axis, the abscissa is always 0.
- At the origin, both abscissa and ordinate are 0.
- The signs of the abscissa and ordinate determine the quadrant of the point.
- Distance from y-axis = |abscissa|; distance from x-axis = |ordinate|.
- The ordered pair (a, b) is NOT the same as (b, a) unless a = b.
- Points with the same abscissa lie on a vertical line; points with the same ordinate lie on a horizontal line.
Practice Problems
- Write the abscissa and ordinate of the points: A(3, −5), B(−2, 7), C(0, 4), D(−1, 0).
- In which quadrant or on which axis do the following points lie: (−3, −8), (5, 0), (0, −6), (2, 9)?
- A point has ordinate −5 and lies on the y-axis. Write its coordinates.
- Find the distance of the point (−9, 12) from both axes.
- Three points have the same ordinate equal to 3. Give possible coordinates and describe their position.
- The reflection of P(a, b) in the y-axis is P′. If P = (−4, 7), find P′.
- A point in Quadrant IV has abscissa 8 and is 5 units from the x-axis. Find the point.
- If the abscissa of a point equals its ordinate, on what line does the point lie?
Frequently Asked Questions
Q1. What is abscissa in coordinate geometry?
The abscissa is the x-coordinate of a point. For point P(x, y), the abscissa is x. It represents the horizontal distance of the point from the y-axis.
Q2. What is ordinate in coordinate geometry?
The ordinate is the y-coordinate of a point. For point P(x, y), the ordinate is y. It represents the vertical distance of the point from the x-axis.
Q3. What is the abscissa of a point on the y-axis?
The abscissa of every point on the y-axis is 0. Such points have the form (0, y).
Q4. What is the ordinate of a point on the x-axis?
The ordinate of every point on the x-axis is 0. Such points have the form (x, 0).
Q5. Why is the order important in an ordered pair?
Because (a, b) and (b, a) refer to different points on the Cartesian plane (unless a = b). The first value is always the abscissa and the second is always the ordinate. Swapping them changes the position.
Q6. What is the abscissa and ordinate of the origin?
Both are 0. The origin has coordinates (0, 0), so abscissa = 0 and ordinate = 0.
Q7. How do you find the distance of a point from the x-axis?
The distance of point P(x, y) from the x-axis is the absolute value of its ordinate: |y|. For example, P(3, −5) is 5 units from the x-axis.
Q8. What happens to the abscissa and ordinate when a point is reflected in the origin?
Both change sign. Point P(a, b) reflected in the origin becomes P′(−a, −b). For example, (3, −7) becomes (−3, 7).
Q9. Can the abscissa and ordinate of a point be equal?
Yes. Points like (2, 2), (−5, −5), and (0, 0) have equal abscissa and ordinate. All such points lie on the line y = x.
Q10. Is abscissa and ordinate in the CBSE Class 9 syllabus?
Yes. These terms are introduced in the CBSE Class 9 chapter on Coordinate Geometry. Students learn to identify abscissa and ordinate, plot points, and determine quadrants.
Related Topics
- Cartesian Plane
- Introduction to Coordinate Geometry
- Plotting Points in Four Quadrants
- Distance Formula
- Section Formula
- Midpoint Formula
- Area of Triangle Using Coordinates
- Centroid of a Triangle
- Collinearity Using Distance Formula
- External Division (Section Formula)
- Coordinate Geometry Word Problems
- Equation of a Line (Introduction)
- Quadrilateral Using Coordinate Geometry
