Coordinate Geometry for CBSE Class 9 Maths (NCERT)

The Cartesian Coordinate System: Your Mathematical Map

At its heart, coordinate geometry provides a way to locate points in a plane using a pair of numbers. This system, known as the Cartesian coordinate system (named after the French mathematician René Descartes), uses two perpendicular number lines that intersect at their zero points. These lines are called axes.

1. The X-axis (Horizontal Axis): This is the horizontal number line. Positive numbers are to the right of the intersection point, and negative numbers are to the left.
2. The Y-axis (Vertical Axis): This is the vertical number line. Positive numbers are above the intersection point, and negative numbers are below.
3. The Origin (O): The point where the X-axis and Y-axis intersect is called the origin. Its coordinates are always (0, 0).

These two axes divide the plane into four regions called quadrants. These are numbered in an anti-clockwise direction starting from the upper right region:

• Quadrant I (QI): (+, +) – Both x and y coordinates are positive.
• Quadrant II (QII): (–, +) – x-coordinate is negative, y-coordinate is positive.
• Quadrant III (QIII): (–, –) – Both x and y coordinates are negative.
• Quadrant IV (QIV): (+, –) – x-coordinate is positive, y-coordinate is negative.

Every point in this plane can be uniquely identified by an ordered pair of numbers (x, y), where 'x' is its horizontal distance from the y-axis (its x-coordinate or abscissa) and 'y' is its vertical distance from the x-axis (its y-coordinate or ordinate). Always remember to write the x-coordinate first, followed by the y-coordinate, enclosed in parentheses. For example, (3, 5) means 3 units along the positive x-axis and 5 units along the positive y-axis.

Key Terms in Coordinate Geometry

Cartesian Coordinate System

A system used to locate points in a plane using two perpendicular number lines (axes) intersecting at the origin.

X-axis

The horizontal number line in the Cartesian system, used to measure the x-coordinate (abscissa).

Y-axis

The vertical number line in the Cartesian system, used to measure the y-coordinate (ordinate).

Origin

The point where the X-axis and Y-axis intersect. Its coordinates are (0, 0).

Abscissa (x-coordinate)

The horizontal distance of a point from the y-axis, indicating its position along the x-axis.

Ordinate (y-coordinate)

The vertical distance of a point from the x-axis, indicating its position along the y-axis.

Coordinates

An ordered pair (x, y) that uniquely identifies the position of a point in the Cartesian plane.

Quadrants

The four regions into which the Cartesian plane is divided by the X-axis and Y-axis.

How to Plot a Point (x, y) on the Cartesian Plane

1. Start at the Origin — Always begin at the origin, which is the point (0, 0).
2. Move along the X-axis (Abscissa) — Look at the x-coordinate (the first number in the pair). If it's positive, move that many units to the right from the origin. If it's negative, move that many units to the left. If it's zero, stay on the y-axis.
3. Move parallel to the Y-axis (Ordinate) — From the position you reached in the previous step, look at the y-coordinate (the second number). If it's positive, move that many units upwards. If it's negative, move that many units downwards. If it's zero, stay on the x-axis.
4. Mark the Point — The final position you reach is the location of your point. Mark it clearly and label it with its coordinates.

Worked Example: Plotting Points

• Example 1: Plot the point P(3, 4). 1. Start at the origin (0, 0). 2. The x-coordinate is 3 (positive), so move 3 units to the right along the x-axis. 3. From there, the y-coordinate is 4 (positive), so move 4 units upwards, parallel to the y-axis. 4. Mark this point and label it P(3, 4). This point lies in Quadrant I.
• Example 2: Plot the point Q(-2, 5). 1. Start at the origin (0, 0). 2. The x-coordinate is -2 (negative), so move 2 units to the left along the x-axis. 3. From there, the y-coordinate is 5 (positive), so move 5 units upwards, parallel to the y-axis. 4. Mark this point and label it Q(-2, 5). This point lies in Quadrant II.
• Example 3: Plot the point R(-4, -1). 1. Start at the origin (0, 0). 2. The x-coordinate is -4 (negative), so move 4 units to the left along the x-axis. 3. From there, the y-coordinate is -1 (negative), so move 1 unit downwards, parallel to the y-axis. 4. Mark this point and label it R(-4, -1). This point lies in Quadrant III.
• Example 4: Plot the point S(5, -3). 1. Start at the origin (0, 0). 2. The x-coordinate is 5 (positive), so move 5 units to the right along the x-axis. 3. From there, the y-coordinate is -3 (negative), so move 3 units downwards, parallel to the y-axis. 4. Mark this point and label it S(5, -3). This point lies in Quadrant IV.
• Example 5: Plot the point T(0, -3). 1. Start at the origin (0, 0). 2. The x-coordinate is 0, so do not move left or right from the origin; stay on the y-axis. 3. From there, the y-coordinate is -3 (negative), so move 3 units downwards along the y-axis. 4. Mark this point and label it T(0, -3). This point lies on the negative Y-axis, not in any quadrant.

Exam Tip: Avoiding Common Mistakes in Coordinate Geometry

Even simple concepts can lead to errors under exam pressure. Here are some common mistakes to watch out for:

1. Confusing Abscissa and Ordinate: Always remember that coordinates are written as (x, y), where x is the abscissa and y is the ordinate. Swapping them will result in plotting a completely different point. For example, (2, 3) is different from (3, 2).
2. Incorrect Movement Along Axes: When plotting, move horizontally (along x-axis) first, then vertically (along y-axis) second. Do not move vertically first unless the x-coordinate is 0.
3. Misidentifying Quadrants: Pay close attention to the signs of the coordinates:

• Quadrant I: (+, +)
• Quadrant II: (–, +)
• Quadrant III: (–, –)
• Quadrant IV: (+, –)

A common error is to number quadrants clockwise instead of anti-clockwise.

1. Points on Axes: Remember that points lying on the x-axis or y-axis are not considered to be in any quadrant. For points on the x-axis, the y-coordinate is 0 (e.g., (5, 0)). For points on the y-axis, the x-coordinate is 0 (e.g., (0, -4)). The origin (0,0) lies on both axes.

Practice Questions with Solutions

• Q: Identify the quadrant or axis for each of the following points: P(2, -3), Q(-4, -1), R(0, 5), S(-2, 6), T(3, 0). A: Step 1: For P(2, -3), the x-coordinate is positive and the y-coordinate is negative. This corresponds to Quadrant IV. Step 2: For Q(-4, -1), both x and y coordinates are negative. This corresponds to Quadrant III. Step 3: For R(0, 5), the x-coordinate is zero. This means the point lies on the y-axis, specifically the positive y-axis. Step 4: For S(-2, 6), the x-coordinate is negative and the y-coordinate is positive. This corresponds to Quadrant II. Step 5: For T(3, 0), the y-coordinate is zero. This means the point lies on the x-axis, specifically the positive x-axis. Final answer: P in QIV, Q in QIII, R on positive Y-axis, S in QII, T on positive X-axis.
• Q: What is the abscissa of the point (-5, 7)? What is its ordinate? A: Step 1: Recall that in an ordered pair (x, y), 'x' is the abscissa and 'y' is the ordinate. Step 2: For the point (-5, 7), the x-coordinate is -5. Step 3: For the point (-5, 7), the y-coordinate is 7. Final answer: The abscissa is -5, and the ordinate is 7.
• Q: A point lies on the x-axis at a distance of 4 units to the left of the origin. Write its coordinates. A: Step 1: Points on the x-axis have their y-coordinate as 0. Step 2: "4 units to the left of the origin" means the x-coordinate is -4. Step 3: Combine these to form the ordered pair. Final answer: The coordinates are (-4, 0).
• Q: Plot the points A(1, 2), B(-3, 2), C(-3, -4), and D(1, -4) on a graph paper and identify the figure formed. A: Step 1: Plot A(1, 2) in QI. Step 2: Plot B(-3, 2) in QII. Step 3: Plot C(-3, -4) in QIII. Step 4: Plot D(1, -4) in QIV. Step 5: Connect the points in order: A to B, B to C, C to D, and D to A. Observe the sides. AB is parallel to CD (both horizontal lines with y=2 and y=-4 respectively), and BC is parallel to DA (both vertical lines with x=-3 and x=1 respectively). Final answer: The figure formed is a rectangle.
• Q: Without plotting, state which axis the point (0, -7) lies on and in which quadrant the point (-10, 2) lies. A: Step 1: For (0, -7), the x-coordinate is 0. Any point with an x-coordinate of 0 lies on the y-axis. Since the y-coordinate is negative, it is on the negative y-axis. Step 2: For (-10, 2), the x-coordinate is negative (-) and the y-coordinate is positive (+). A point with (–, +) coordinates lies in Quadrant II. Final answer: (0, -7) lies on the Y-axis. (-10, 2) lies in Quadrant II.

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