Coordinate Geometry
CBSE Class 9 · Mathematics · Notes and practice questions
Class 9 Coordinate Geometry builds the Cartesian plane from two perpendicular number lines, and shows how to describe any point uniquely by an ordered pair of numbers. It covers the axes, quadrants, abscissa and ordinate, and the plotting and reading of coordinates.
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Coordinate geometry starts from the Cartesian plane, a flat surface with two perpendicular number lines crossing at their zero marks. The horizontal line is the x-axis and the vertical line is the y-axis; their intersection is the origin O. The two axes make it possible to describe a point by a pair of real numbers. Those two numbers are written in a fixed order, and the chapter is mainly about understanding what that order means and how it locates a point.
The first number in the pair is the x-coordinate or abscissa and the second is the y-coordinate or ordinate. The abscissa tells the signed distance from the y-axis, so positive values are to the right and negative values to the left. The ordinate tells the signed distance from the x-axis, with positive values above and negative values below. Together they give the precise location of a point, and no two different points share the same ordered pair.
The two axes divide the plane into four regions called quadrants, usually numbered anticlockwise starting from the top right. In the first quadrant both coordinates are positive; in the second x is negative and y is positive; in the third both are negative; in the fourth x is positive and y is negative. Points that lie directly on an axis have one coordinate equal to zero, so they are not inside any quadrant. Recognising these sign patterns is the most frequently tested skill in this chapter.
To plot a point such as (-3,2), start at the origin, move 3 units left along the x-axis because the abscissa is negative, then move 2 units up parallel to the y-axis. To read the coordinates of a plotted point, drop perpendicular lines to the axes and read the scale markers. This chapter introduces no length or division formulas; success comes from practising positions, signs and the standard order of writing coordinates. Care with ordered pairs and sign conventions prevents the majority of classroom mistakes.
Key terms
- Cartesian plane
- A coordinate plane formed by two perpendicular number lines, the x-axis and the y-axis, crossing at the origin. Every point in it is identified by an ordered pair (x, y).
- x-axis
- The horizontal number line in the Cartesian plane. Points to the right of the origin have positive x-coordinates; points to the left have negative x-coordinates.
- y-axis
- The vertical number line in the Cartesian plane. Points above the origin have positive y-coordinates; points below have negative y-coordinates.
- Origin
- The point of intersection of the x-axis and y-axis. It has coordinates (0, 0) and acts as the zero point from which all positions are measured.
- Ordered pair
- The pair of coordinates (x, y) that fixes the position of a point in the plane. The x-coordinate is written first and the y-coordinate second; swapping them usually gives a different point.
- Abscissa
- The x-coordinate of a point. It gives the signed distance from the y-axis: positive to the right, negative to the left.
- Ordinate
- The y-coordinate of a point. It gives the signed distance from the x-axis: positive above, negative below.
- Quadrant
- One of the four regions into which the x-axis and y-axis divide the plane. Quadrants are numbered anticlockwise beginning from the upper-right region.
- Point on an axis
- A point on the x-axis has coordinates (a, 0), and a point on the y-axis has coordinates (0, b). Such a point lies on an axis rather than in any quadrant.
Practice questions with answers
1. What are the coordinates of the point 5 units to the right of the y-axis and 3 units above the x-axis?
The point is 5 units to the right, so its x-coordinate is 5; it is 3 units above, so its y-coordinate is 3. The coordinates are (5, 3).
2. A point lies 4 units to the left of the y-axis and on the x-axis. Write its coordinates and name its location.
To the left of the y-axis gives a negative x-coordinate, so x = −4; on the x-axis gives y = 0. The coordinates are (−4, 0), and the point lies on the x-axis, not in any quadrant.
3. If the abscissa of a point is −2 and the ordinate is 6, in which quadrant does the point lie?
The coordinates are (−2, 6). Since x is negative and y is positive, the point lies in the second quadrant.
4. Write the signs of the x-coordinate and y-coordinate in each of the four quadrants.
Quadrant I: (+, +); Quadrant II: (−, +); Quadrant III: (−, −); Quadrant IV: (+, −). Points on an axis have one coordinate zero and are not in any quadrant.
5. State whether (0, −9) lies on an axis or in a quadrant, and justify your answer.
The x-coordinate is 0, so the point lies on the y-axis, 9 units below the origin. A point with one coordinate zero belongs to an axis, not to a quadrant.
6. Starting at the origin, how would you locate the point (3, −4) on a Cartesian plane?
Move 3 units to the right along the x-axis because x = 3, then move 4 units down because y = −4. Mark the final position; it lies in the fourth quadrant.
7. The coordinates of two points are (4, 7) and (7, 4). Explain why the order of the numbers matters.
In (4, 7), x = 4 and y = 7, meaning 4 units right and 7 units up. In (7, 4), x = 7 and y = 4, meaning 7 units right and 4 units up. These are different points, so the x-coordinate must always be written first.
8. What is the name of the point of intersection of the x-axis and the y-axis, and what are its coordinates?
This point is the origin, denoted by O. Its coordinates are (0, 0), because it has zero distance from both axes.
9. Define the abscissa and ordinate of a point with coordinates (a, b).
The abscissa is a, the x-coordinate, which is the signed distance from the y-axis. The ordinate is b, the y-coordinate, which is the signed distance from the x-axis.
10. Write the coordinates of the point 2 units to the right of the y-axis and 3 units below the x-axis, and state its quadrant.
2 units to the right gives x = 2, and 3 units below gives y = −3. The coordinates are (2, −3). Since x is positive and y is negative, the point lies in the fourth quadrant.
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