Co Ordinate Geometry Class 9 ICSE — Maths Chapter 26
Co ordinate geometry class 9 ICSE (Chapter 26 of the Mathematics textbook) introduces the branch of mathematics that fixes the position of a point using an ordered pair of numbers called co-ordinates. These co-ordinates are measured relative to two perpendicular number lines, the co-ordinate axes. The chapter then moves from naming variables to plotting points, graphing linear equations, and reading slope and y-intercept from a line’s equation.
ICSE Class 9 Mathematics Chapter 26 Co Ordinate Geometry PDF
What is inside Chapter 26
13 pages · 12 sections · 9 worked examples · 23 exercise questions
| Section | Title |
|---|---|
| 26.1 | INTRODUCTION |
| 26.2 | DEPENDENT AND INDEPENDENT VARIABLES |
| 26.3 | ORDERED PAIR |
| 26.4 | CARTESIAN PLANE |
| 26.5 | CO-ORDINATES OF POINTS |
| 26.6 | QUADRANTS AND SIGN CONVENTION |
| 26.7 | PLOTTING OF POINTS |
| 26.8 | GRAPHS OF x = 0, y = 0, x = a, y = a, etc. |
| 26.9 | GRAPHING A LINEAR EQUATION |
| 26.10 | INCLINATION AND SLOPE |
| 26.11 | Y-INTERCEPT |
| 26.12 | FINDING THE SLOPE AND THE Y-INTERCEPT OF A GIVEN LINE |
- EXERCISE 26(A): 13 questions
- EXERCISE 26(C): 10 questions
26.1 INTRODUCTION
Co-ordinate geometry is the branch of mathematics in which a pair of numbers, called co-ordinates, is used to represent the position of a point. The position is measured with respect to two mutually perpendicular number lines, called the co-ordinate axes.
The location of points falls under the heading co-ordinate. Their relations with different geometrical figures fall under geometry. Together, locating points and relating them to figures is what co-ordinate geometry studies. That is why the chapter connects plotting with shapes like rectangles, squares, and straight lines.
This one definition drives everything that follows. Ordered pairs fix the position. Quadrants and sign conventions tell you where a point sits. Graphing turns an equation into a picture. If you hold on to “co-ordinate means location” and “geometry means shapes,” each later section builds naturally on this one.
The topics follow the pattern set out for this level by the CISCE syllabus, and you can see how the chapter fits with the rest of the Class 9 Mathematics chapters as you work through them.
26.2 DEPENDENT AND INDEPENDENT VARIABLES
In linear equations, the letters x and y are called variables. Whether a variable is dependent or independent depends on which letter is the subject of the formula.
If y is the subject, as in y = 4x + 9, then y is the dependent variable and x is the independent variable. The value of y depends on the value of x. If x is the subject, as in x = 3y - 5, then x is dependent and y is independent.
Worked pattern: take 3x - 2y + 6 = 0. To make x the subject: 3x = 2y - 6, so x = \frac{2}{3}y - 2. Here x is dependent and y is independent. To make y the subject: -2y = -3x - 6, so 2y = 3x + 6, giving y = \frac{3}{2}x + 3. Now y is dependent and x is independent.
Exercise 26(A) opens by asking you to name the dependent and independent variable in each given equation. Look at which letter stands alone on one side of the equals sign. That letter is the dependent one.
26.3 ORDERED PAIR
An ordered pair is a pair of two numbers written in a specific order. The order is important and necessary. (5, 7) and (7, 5) are different pairs because the components swap places.
To form an ordered pair, write the numbers in order, separate them with a comma, and enclose them in brackets. In (a, b), a is the first component and b is the second component.
The equality rule: (a, b) = (c, d) means a = c and b = d. Both components must match separately. An ordered pair may also have both components equal, like (0, 0) or (5, 5).
Worked pattern: find x and y if (2x - 1, 3y + 2) = (7, 8). Match first components: 2x - 1 = 7, so 2x = 8 and x = 4. Match second components: 3y + 2 = 8, so 3y = 6 and y = 2.
This rule turns one equation into two separate equations. That is exactly what Exercise 26(A) question 3 uses to find x and y.
26.4 CARTESIAN PLANE
A cartesian plane consists of two mutually perpendicular number lines intersecting at their zeros. It is also called a co-ordinate plane.
The horizontal number line \mathrm{XOX'} is called the x-axis. The vertical number line \mathrm{YOY'} is called the y-axis. Their point of intersection O is the origin, which is zero for both axes.
The system formed by the x-axis, the y-axis, and the origin together is the cartesian co-ordinate system. The two axes together are called the co-ordinate axes.
Every point in this chapter is measured from the origin. The origin is your fixed starting point for plotting, for reading co-ordinates, and for describing where a line meets an axis. Because both axes pass through zero at O, every distance along either axis is measured from that same reference point.
26.5 CO-ORDINATES OF POINTS
The position of each point in the plane is fixed by an ordered pair measured from the origin.
The x-co-ordinate, or abscissa, is the distance of the point along the x-axis. The y-co-ordinate, or ordinate, is the distance along the y-axis.
Co-ordinates are always written as (abscissa, ordinate). The abscissa comes first, then the ordinate, separated by a comma and enclosed in brackets. For the point (-3, 5), the abscissa is -3 and the ordinate is 5.
If a point has co-ordinates (x, y), then x is the abscissa and y is the ordinate. The order is fixed by convention. Writing the ordinate first, or reading co-ordinates backwards, plots the wrong point entirely. The sign tells you the direction of each distance: negative abscissa means left of the origin, negative ordinate means below it.
26.6 QUADRANTS AND SIGN CONVENTION
The co-ordinate axes divide the plane into four parts called quadrants. Starting from OX and moving anti-clockwise: XOY is the first quadrant, YOX’ the second, X’OY’ the third, and Y’OX the fourth.
Each quadrant has a fixed sign convention for its co-ordinates:
| Quadrant | Name | Signs |
|---|---|---|
| First | XOY | (+, +) |
| Second | X’OY | (-, +) |
| Third | X’OY’ | (-, -) |
| Fourth | Y’OX | (+, -) |
To remember the signs in order, use the mnemonic All Students Take Coffee. The first letters map to the four quadrants: (+, +), (-, +), (-, -), (+, -).
Edge case: the origin and any point on either axis lie in no quadrant. A point like (3, 0) sits on the x-axis, not inside a quadrant.
Common mistakes to avoid:
- Confusing abscissa and ordinate. The abscissa is the x-distance; the ordinate is the y-distance.
- Mixing up quadrant signs, especially the second quadrant (-, +) and the fourth quadrant (+, -).
- Placing an on-axis point inside a quadrant. Points on the axes belong to no quadrant.
26.7 PLOTTING OF POINTS
Plotting a point takes two steps, always starting from the origin O.
Step 1: move along the x-axis by the abscissa. Positive means right, negative means left. Step 2: from there, move parallel to the y-axis by the ordinate. Positive means up, negative means down. Place a dot and label the point.
Three special forms are worth memorising:
- The origin is (0, 0).
- A point on the x-axis has the form (x, 0); its ordinate is zero.
- A point on the y-axis has the form (0, y); its abscissa is zero.
Worked pattern: plot A(3, 4), B(-2, 5), C(-4, -3), and D(5, -2). For A, move 3 right, then 4 up. For B, move 2 left, then 5 up. For C, move 4 left, then 3 down. For D, move 5 right, then 2 down.
Sometimes the co-ordinates come from solving equations. Worked pattern: find the point whose abscissa solves 2x + 1 = 7 and whose ordinate solves 3y - 2 = 10. First: 2x = 6, so x = 3. Second: 3y = 12, so y = 4. The point is (3, 4).
Exercise 26(A) also asks you to find the fourth vertex of a rectangle or square. Plot the three given vertices, complete the shape, and read the fourth point from the graph.
26.8 GRAPHS OF x = 0, y = 0, x = a, y = a, etc.
Four rules cover the axes and lines parallel to them.
x = 0 is the equation of the y-axis, because every point on the y-axis has abscissa 0. y = 0 is the equation of the x-axis, because every point on the x-axis has ordinate 0.
x = a is the equation of a line parallel to the y-axis at a distance of a units from it. y = a is the equation of a line parallel to the x-axis at a distance of a units from it.
Quick example: x = -2 is a vertical line 2 units left of the y-axis. y = 5 is a horizontal line 5 units above the x-axis.
| Form | Direction | Slope |
|---|---|---|
| x = a | Vertical, parallel to y-axis | Not defined |
| y = a | Horizontal, parallel to x-axis | 0 |
Common mistake: x = 4 is vertical, not horizontal. Fixing x pins the horizontal co-ordinate, so the line runs up and down. Fixing y pins the vertical co-ordinate, so the line runs left and right.
26.9 GRAPHING A LINEAR EQUATION
An equation whose graph is a straight line is a linear equation. To draw its graph, plot a few points that satisfy the equation, then draw a straight line through those points.
The three-step table method:
- Choose at least three suitable values of x.
- Compute the matching values of y.
- Build a table, plot the points, and draw the line through them.
The chapter covers two types. Type 1 is y = mx, where the line passes through the origin. Type 2 is y = mx + c, where c is a rational number but not zero.
Worked pattern: graph y = 2x - 1. Pick x = -1, 0, and 2. For x = -1, y = 2(-1) - 1 = -3, giving (-1, -3). For x = 0, y = -1, giving (0, -1). For x = 2, y = 4 - 1 = 3, giving (2, 3).
Plot the three points and join them with a straight line. Three points are safer than two, because a third point lets you check that all three lie on one line.
After drawing, you can read values back from the graph. To find x for a given y, draw a horizontal line from that y to the graph, then a vertical line down to the x-axis. To find y for a given x, do the reverse. This reading-back skill appears in the exercises later in the chapter.
26.10 INCLINATION AND SLOPE
Inclination \theta is the angle a straight line makes with the positive direction of the x-axis, measured anti-clockwise. The slope (or gradient) m of the line is \tan \theta.
m = \tan \theta
Two fixed cases follow. The x-axis and every line parallel to it have inclination \theta = 0^\circ, so slope m = \tan 0^\circ = 0. The y-axis and every line parallel to it have inclination \theta = 90^\circ, and the slope is not defined.
Standard slopes for common angles:
| Inclination \theta | Slope m = \tan \theta |
|---|---|
| 0^\circ | 0 |
| 30^\circ | \frac{1}{\sqrt{3}} |
| 45^\circ | 1 |
| 60^\circ | \sqrt{3} |
Why is a vertical line’s slope undefined? \tan 90^\circ has no finite value; the tangent grows without bound near 90^\circ. So a vertical line has no number to call its slope.
This connects to section 26.8. A horizontal line y = a has slope 0, matching \theta = 0^\circ. A vertical line x = a has no slope, matching \theta = 90^\circ.
26.11 Y-INTERCEPT
The y-intercept c is the distance from the origin to the point where a line meets the y-axis. It is usually denoted by c.
The sign depends on where the line crosses. The y-intercept is positive if measured above the origin, and negative if measured below the origin.
Two zero cases: the x-axis itself has y-intercept 0, and every line parallel to the y-axis has y-intercept 0. A vertical line x = a never meets the y-axis, so it has no intercept to measure.
Illustration: a line crossing the y-axis at (0, -3) has y-intercept -3. The point sits 3 units below the origin, so the intercept carries a negative sign.
The y-intercept alone does not describe a line fully. It only tells you where the line cuts the y-axis. In a graph, you read it where the line crosses the vertical axis, and the next section pairs it with slope to give the complete equation.
26.12 FINDING THE SLOPE AND THE Y-INTERCEPT OF A GIVEN LINE
The standard method for any line written as ax + by + c = 0 is to make y the subject. Then read the slope and the y-intercept from the rearranged form.
From ax + by + c = 0: by = -ax - c, so y = -\frac{a}{b}x - \frac{c}{b}. The coefficient of x is the slope m = -\frac{a}{b}. The constant term is the y-intercept c = -\frac{c}{b}.
Worked pattern: for 3x + 2y - 7 = 0, rearrange to get 2y = -3x + 7, then y = -\frac{3}{2}x + \frac{7}{2}. So slope m = -\frac{3}{2} and y-intercept = \frac{7}{2}.
The converse rule: if the slope is m and the y-intercept is c, the equation of the line is y = mx + c. Worked pattern: a line with slope 4 and y-intercept -5 has equation y = 4x - 5.
Why does this work? Slope and y-intercept together fix which straight line you get. Every point on the line satisfies y = mx + c. Exercise 26(C) asks you to apply both directions: read slope and intercept from an equation, and build an equation from given slope and intercept. This method also appears in other Class 9 Mathematics chapters that follow this one.
What to practise from this chapter
- EXERCISE 26(A) — 13 questions on dependent and independent variables, plotting points, ordered pairs, reading co-ordinates from a graph, and finding the fourth vertex of rectangles, squares, and parallelograms.
- EXERCISE 26(C) — 10 questions on inclination, slope, y-intercept, and drawing straight lines from equations.
These two exercises cover the three core skills the board tests from this chapter: plotting or reading points, finding slope and y-intercept from an equation, and drawing a straight line from its equation.
Frequently asked questions
What is the difference between the abscissa and the ordinate of a point?
The abscissa is the x-co-ordinate, the distance of the point along the x-axis. The ordinate is the y-co-ordinate, the distance along the y-axis. Co-ordinates are always written (abscissa, ordinate).
How do I find the slope and y-intercept of a line from ax + by + c = 0?
Make y the subject of the equation. From ax + by + c = 0, you get y = -\frac{a}{b}x - \frac{c}{b}. The coefficient of x is the slope, and the constant term is the y-intercept.
Why is the slope of a horizontal line zero but the slope of a vertical line not defined?
A horizontal line has inclination \theta = 0^\circ, and \tan 0^\circ = 0, so its slope is 0. A vertical line has inclination \theta = 90^\circ, and \tan 90^\circ has no finite value, so its slope is not defined.
In which quadrant does a point with a negative abscissa and a positive ordinate lie?
Negative abscissa and positive ordinate means the sign pair (-, +). That sign pair belongs to the second quadrant, YOX’.
What is the equation of a line parallel to the y-axis at a distance of 4 units from it?
A line parallel to the y-axis at a distance of a units has equation x = a. At a distance of 4 units, the equation is x = 4.
Reference: ICSE Class 9 Mathematics textbook material, chapter 26: Co Ordinate Geometry — CISCE-aligned.