ICSE Class 9 Maths Chapter 26 Co ordinate Geometry

Unit 8: Co-ordinate Geometry

26.1 Introduction

Co-ordinate Geometry is the branch of mathematics in which a pair of two numbers, called co-ordinates, is used to represent the position of a point with respect to two mutually perpendicular number lines called co-ordinate axes.

The location of points comes under the heading co-ordinate and their relations, with respect to different figures, come under the heading geometry.

Together, the location of the points and their relationship with different geometrical figures is called Co-ordinate Geometry.

26.2 Dependent And Independent Variables

In linear equations of the form: \(3x + 4y = 5\), \(x - 3y + 8 = 0\), \(y = mx + c\), \(x = 5y - 8\), etc., the letters 'x' and 'y' are called variables.

1. If a linear equation in x and y is expressed with y as the subject of formula (equation); y is called the dependent variable and x is called the independent variable. In each of the following equations; y is dependent variable and x is independent variable.

(i) \(y = 3x - 6\)

(ii) \(y = 5 - \frac{x}{4}\)

(iii) \(y = 2(3x - 5) + 7\)

2. If a linear equation in x and y is expressed with x as the subject of formula (equation); x is called the dependent variable and y is called the independent variable. In each of the following equations; x is the dependent variable and y is the independent variable.

(i) \(x = 5y + 7\)

(ii) \(x = 5(5y + 8) - 10\)

(iii) \(x = 7 - \frac{2y}{3}\)

In equation \(y = 4x + 9\); the value of y depends on the value of x, so y is said to be dependent variable and x is said to be independent variable.

In the same way, in equation \(x = 3y - 5\); the value of x depends on the value of y, so x is said to be dependent variable and y is said to be independent variable.

Example 1

Express the equation \(4x - 5y + 20 = 0\) in the form so that:

(i) x is dependent variable and y is independent variable.

(ii) y is dependent variable and x is independent variable.

Solution

(i) \(4x - 5y + 20 = 0 \Rightarrow 4x = 5y - 20\)

\(\Rightarrow x = \frac{5}{4}y - 5\)

(ii) \(4x - 5y + 20 = 0 \Rightarrow -5y = -4x - 20\)

\(\Rightarrow 5y = 4x + 20\)

\(\Rightarrow y = \frac{4}{5}x + 4\)

26.3 Ordered Pair

An ordered pair means, a pair of two objects taken in a specific order. In relation to co-ordinate geometry, an ordered pair means, a pair of two numbers in which the order is important and necessary.

1. To form an ordered pair, the numbers are written in specific order, separated by a comma, and enclosed in small brackets.

Each of the following represents an ordered pair:

\((5, 7), (-6, 8), (0, 0), (0, -6), (5, 0), (3\frac{1}{2}, -2)\), etc.

2. In the ordered pair (a, b); a is called its first component and b is called its second component.

3. Ordered pairs (5, 7) and (7, 5) are different i.e. (5, 7) ≠ (7, 5).

4. If two ordered pairs are equal; their corresponding components are equal i.e. (a, b) = (c, d) ⇒ a = c and b = d.

5. An ordered pair can have both of its components equal i.e. an ordered pair can be of the form: (5, 5), (-6, -6), (0, 0), etc.

Example 2

Find the values of x and y, if:

(i) (x, 4) = (-7, y)

(ii) (x - 3, 6) = (4, x + y)

Solution

Two ordered pairs are equal ⇒ Their first components are equal and their second components are separately equal.

(i) (x, 4) = (-7, y)

⇒ \(x = -7\) and \(y = 4\)

(ii) (x - 3, 6) = (4, x + y)

⇒ \(x - 3 = 4\) and \(6 = x + y\)

⇒ \(x = 7\) and \(6 = 7 + y\)

or \(x = 7\) and \(y = -1\)

26.4 Cartesian Plane

A cartesian (or a co-ordinate) plane consists of two mutually perpendicular number lines intersecting each other at their zeros.

The adjoining figure shows a cartesian plane consisting of two mutually perpendicular number lines XOX' and YOY' intersecting each other at their zero O.

1. The horizontal number line XOX' is called the x-axis.

2. The vertical number line YOY' is called the y-axis.

3. The point of intersection 'O' is called the origin which is zero for both the axes.

The system consisting of the x-axis, the y-axis and the origin is also called cartesian co-ordinate system. The x-axis and the y-axis together are called co-ordinate axes.

26.5 Co-Ordinates Of Points

The position of each point in a co-ordinate plane is determined by means of an ordered pair (a pair of numbers) with reference to the co-ordinate axes as stated below:

(i) Starting from the origin O, measure the distance of the point along x-axis. This distance is called x-co-ordinate or abscissa of the point.

(ii) Starting from the origin O, measure the distance of the point along the y-axis. This distance is called the y-co-ordinate or ordinate of the point.

Thus, the co-ordinates of the point = Position of the point with reference to co-ordinate axes. = (abscissa, ordinate).

In stating the co-ordinates of a point, the abscissa precedes the ordinate and both are enclosed in a small bracket after being separated by a comma.

e.g. if the abscissa of a point is x and its ordinate is y, its co-ordinates = (x, y).

26.6 Quadrants And Sign Convention

1. Quadrants

As shown in the adjoining diagram, the co-ordinate axes divide a co-ordinate plane into four parts, which are known as quadrants. Each point in the plane is located either in one of the quadrants or on one of the axes.

Starting from OX in the anti-clockwise direction; XOY is called the first quadrant, XOY' is called the second quadrant, X'OY' is called the third quadrant and Y'OX is called the fourth quadrant.

2. Sign Convention

It is clear from the figure (given on the previous page); the co-ordinate axes divide a plane into four quadrants. Also:

(i) in the first quadrant, XOY, the abscissa and the ordinate both are positive

(ii) in the second quadrant, X'OY, the abscissa is negative and the ordinate is positive

(iii) in the third quadrant, X'OY', the abscissa and the ordinate both are negative and

(iv) in the fourth quadrant, XOY'; the abscissa is positive and the ordinate is negative.

26.7 Plotting Of Points

Example 3

Plot the points A (4, 2), B (-5, 3), C (-4, -5) and D (5, -2).

Solution

On a graph paper, draw the co-ordinate axes XOX' and YOY' intersecting at origin O. With proper scale, mark the numbers on the two co-ordinate axes.

For plotting any point; two steps are to be adopted, e.g. to plot point A (4, 2).

Step 1

Starting from the origin O, move 4 units along the positive direction of the x-axis i.e. to the right of the origin O.

Step 2

Now, from there, move 2 units up (i.e. parallel to positive direction of the y-axis) and place a dot at the point reached. Label this point as A (4, 2).

Similarly, plot the other points B (-5, 3), C (-4, -5) and D (5, -2)

1. The co-ordinates of the origin = (0,0)

2. For a point on the x-axis, its ordinate is always zero and so the co-ordinates of a point on x-axis is of the form (x, 0).

e.g. (7, 0), (3, 0), (0, 0), (-4, 0), (-8, 0), etc.

3. For a point on the y-axis; its abscissa is always zero and so the co-ordinates of a point on y-axis is of the form (0, y).

e.g. (0, 8), (0, 3), (0, 0), (0, -2), (0, -5), etc.

Example 4

A (3, 6), B (3, 2) and C (8, 2) are the vertices of a rectangle. Plot these points on a graph paper and then use it to find the co-ordinates of the vertex D.

Solution

After plotting the given points A, B and C on a graph paper; join A with B and B with C.

Complete the rectangle ABCD.

Now, read the co-ordinates of D.

As is clear from the graph; D = (8, 6)

Example 5

Find the co-ordinates of the point whose abscissa is the solution of the first quadrant and the ordinate is the solution of the second equation.

0-5x - 3 = -0.25x and 8 - 0-2 (y + 3) = 3y + 1

Solution

\(0 \cdot 5x - 3 = -0 \cdot 25x \Rightarrow 0 \cdot 5x + 0 \cdot 25x = 3\)

\(\Rightarrow 0 \cdot 75x = 3\)

\(\Rightarrow x = \frac{3}{0 \cdot 75} = \frac{3 \times 100}{75} = 4\)

\(8 - 0 \cdot 2(y + 3) = 3y + 1 \Rightarrow 8 - 0 \cdot 2y - 0 \cdot 6 = 3y + 1\)

\(\Rightarrow -0 \cdot 2y - 3y = 1 + 0 \cdot 6 - 8\)

\(\Rightarrow -3 \cdot 2y = -6 \cdot 4 \Rightarrow y = 2\)

The co-ordinates of the point = (4, 2)

Teacher's Note

Coordinate geometry helps us understand locations on maps and GPS systems, which we use in everyday navigation and delivery services.

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