CBSE Class 9 Maths: Coordinate Geometry Exercise 3.2 Explained

Understanding the Cartesian Plane and Quadrants

The Cartesian plane, also known as the coordinate plane, is formed by two perpendicular number lines that intersect at their zero points. These lines are called the coordinate axes. The horizontal line is the x-axis (or abscissa axis), and the vertical line is the y-axis (or ordinate axis). Their point of intersection is called the origin, usually denoted by O (0,0).

These two axes divide the plane into four regions, called quadrants. Each quadrant has a specific sign convention for the coordinates (x, y) of any point lying within it:

1. First Quadrant (Q1): Points in this quadrant have both x and y coordinates positive. (x > 0, y > 0). Example: (2, 3).
2. Second Quadrant (Q2): Points in this quadrant have a negative x-coordinate and a positive y-coordinate. (x < 0, y > 0). Example: (-4, 1).
3. Third Quadrant (Q3): Points in this quadrant have both x and y coordinates negative. (x < 0, y < 0). Example: (-5, -2).
4. Fourth Quadrant (Q4): Points in this quadrant have a positive x-coordinate and a negative y-coordinate. (x > 0, y < 0). Example: (6, -7).

Points lying on the x-axis have their y-coordinate as 0 (e.g., (3, 0), (-2, 0)). Points lying on the y-axis have their x-coordinate as 0 (e.g., (0, 5), (0, -1)). The origin (0,0) lies on both axes and is the intersection point.

Key Terms in Coordinate Geometry

Coordinate Axes

The two perpendicular number lines (x-axis and y-axis) that intersect at the origin to form the Cartesian plane.

Origin

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

Abscissa

The x-coordinate of a point. It represents the perpendicular distance of the point from the y-axis.

Ordinate

The y-coordinate of a point. It represents the perpendicular distance of the point from the x-axis.

Coordinates of a Point

An ordered pair (x, y) that uniquely identifies the position of a point on the Cartesian plane, where 'x' is the abscissa and 'y' is the ordinate.

Quadrant

One of the four regions into which the coordinate axes divide the Cartesian plane.

Step-by-Step Guide to Plotting Points

1. Understand the Coordinates — Every point is represented by an ordered pair (x, y), where 'x' is the horizontal distance from the origin (along the x-axis) and 'y' is the vertical distance from the origin (along the y-axis).
2. Start at the Origin (0,0) — Always begin your plotting journey from the origin, which is the intersection of the x-axis and y-axis.
3. Move Horizontally (x-coordinate) — Look at the x-coordinate. If 'x' is positive, move 'x' units to the right from the origin along the x-axis. If 'x' is negative, move 'x' units to the left. If 'x' is zero, stay on the y-axis.
4. Move Vertically (y-coordinate) — From your current position after moving horizontally, look at the y-coordinate. If 'y' is positive, move 'y' units upwards, parallel to the y-axis. If 'y' is negative, move 'y' units downwards. If 'y' is zero, stay on the x-axis.
5. Mark the Point — The final position you reach is the location of your point. Mark it clearly and label it with its coordinates, e.g., P(x, y). Example: To plot the point A(3, -2): 1. Start at (0,0). 2. x-coordinate is 3 (positive), so move 3 units to the right along the x-axis. 3. y-coordinate is -2 (negative), so from the position (3,0), move 2 units downwards. 4. Mark this point as A(3, -2). This point lies in the Fourth Quadrant.

Exam Tips and Common Mistakes to Avoid

To ace your exams in Coordinate Geometry, keep these pointers in mind and avoid common pitfalls:

1. Don't Mix Up (x, y) and (y, x): The order matters! Always remember that the first coordinate is 'x' (horizontal movement) and the second is 'y' (vertical movement). Plotting (2, 3) is different from plotting (3, 2).
2. Quadrant Signs: Carefully check the signs of the coordinates to determine the correct quadrant. A positive x and negative y means Q4, not Q1 or Q3.
3. Points on Axes: Understand that any point on the x-axis has a y-coordinate of 0 (e.g., (5, 0)), and any point on the y-axis has an x-coordinate of 0 (e.g., (0, -4)). These points do not lie in any quadrant but on the axes.
4. Use a Scale: When plotting points on graph paper, always choose an appropriate scale and label your axes clearly (X, X', Y, Y', and the origin O). This ensures accuracy.
5. Practice Reading Coordinates: Sometimes, questions ask you to identify coordinates from a given graph. Practice reading the x-value first by dropping a perpendicular to the x-axis, then the y-value by dropping a perpendicular to the y-axis.

Practice Questions with Solutions

• Q: In which quadrant or on which axis do each of the points (-2, 4), (3, -1), (-1, 0), (1, 2), and (-3, -5) lie? A: Step 1: Analyze (-2, 4). The x-coordinate is negative, and the y-coordinate is positive. This combination corresponds to the Second Quadrant. Step 2: Analyze (3, -1). The x-coordinate is positive, and the y-coordinate is negative. This corresponds to the Fourth Quadrant. Step 3: Analyze (-1, 0). The y-coordinate is 0. Any point with a y-coordinate of 0 lies on the x-axis. Since x is negative, it lies on the negative x-axis. Step 4: Analyze (1, 2). Both x and y coordinates are positive. This corresponds to the First Quadrant. Step 5: Analyze (-3, -5). Both x and y coordinates are negative. This corresponds to the Third Quadrant. Final answer: (-2, 4) in Q2; (3, -1) in Q4; (-1, 0) on negative x-axis; (1, 2) in Q1; (-3, -5) in Q3.
• Q: Plot the points (5, 0), (0, 4), (-3, 0), and (0, -2) on a Cartesian plane and state where they lie. A: Step 1: Draw the Cartesian plane with labeled x and y axes and the origin (0,0). Step 2: For (5, 0): Move 5 units to the right from the origin along the x-axis. Since y=0, it lies on the x-axis. Plot the point. Step 3: For (0, 4): Move 4 units upwards from the origin along the y-axis. Since x=0, it lies on the y-axis. Plot the point. Step 4: For (-3, 0): Move 3 units to the left from the origin along the x-axis. Since y=0, it lies on the x-axis. Plot the point. Step 5: For (0, -2): Move 2 units downwards from the origin along the y-axis. Since x=0, it lies on the y-axis. Plot the point. Final answer: All these points lie on the coordinate axes. (5,0) on positive x-axis; (0,4) on positive y-axis; (-3,0) on negative x-axis; (0,-2) on negative y-axis.
• Q: What are the coordinates of the origin? Does it lie in any quadrant? A: Step 1: Recall the definition of the origin. It is the intersection point of the x-axis and y-axis. Step 2: By convention, this point is assigned the coordinates where both x and y are zero. Step 3: Consider the definition of quadrants. Quadrants are regions between the axes, where coordinates are strictly positive or negative (not zero). Final answer: The coordinates of the origin are (0, 0). It does not lie in any quadrant; it lies on both the x-axis and y-axis.
• Q: A point's abscissa is -5 and its ordinate is 3. Write its coordinates and identify the quadrant it belongs to. A: Step 1: Understand that 'abscissa' is the x-coordinate and 'ordinate' is the y-coordinate. So, x = -5 and y = 3. Step 2: Form the ordered pair (x, y) = (-5, 3). Step 3: Analyze the signs: x is negative, y is positive. A negative x-coordinate and a positive y-coordinate place the point in the Second Quadrant. Final answer: The coordinates of the point are (-5, 3), and it lies in the Second Quadrant.

Frequently Asked Questions