**Introduction**

The French Mathematician and Philosopher Rene Descartes first published his book La Geometric in 1637 in which he used algebra in the study of geometry. This he did by representing points in a plane by ordered pairs of real numbers, called Cartesian coordinates ( named after Rene Descartes ).

Let us now learn about the Cartesian coordinates system.

**The Cartesian Coordinate System**

A Cartesian coordinate plane is a system that uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric elements. This plane is determined by two perpendicular lines, called the x-axis and the y-axis. Let us learn more about these lines.

Let X ‘ O X and Y ‘ O Y be two mutually perpendicular lines through a point O in the plane of a graph paper as shown below.

The line X O X ‘ is called the x-axis or axis of x and the line Y O Y ‘ is called the y-axis or the axis of y. The two lines X O X ‘ and Y O Y ‘ taken together are called the coordinate axis or the axis of the coordinate system. The point O is called the point of origin. This can also be represented as –

**Quadrants of the Coordinate System**

The x-axis and the y-axis together divide the entire coordinate system into four equal parts which are called the quadrants of the coordinate system.

For example, in the graph of the coordinate axis that we have defined above, the two lines X O X ‘ and Y O Y ‘ divide the plane of the graph paper into four regions, namely, X O Y, X ‘ O Y , X ‘ O Y ‘ and Y ‘ O X. These four regions are called the quadrants.

The ray O X ‘ is taken as a positive x-axis and the O X is taken as negative x-axis. Similarly, the ray O Y is taken as a positive y-axis and the O y ‘ is taken as negative y-axis. The signs of the x coordinate and the y coordinate in the four quadrants are thus defined as –

1 st Quatarant – x > 0 and y > 0

2 nd Quadrant – x < 0 and y > 0

3 rd Quarant – x < 0 and y < 0

4 th Quadrant – x > 0 and y < 0

Graphically, the values and x and y in the four quadrants can be represented as –

Now let us see how to identify the coordinates of y in accordance with the coordinates of x on the x –y axis.

**Identifying Y Coordinates**

In coordinate geometry, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, to write the ordered pair using the correct order

Let us understand this by an example.

Suppose we have the following point on the x-y axis.

Let us check the plotting of the points A, B, C and D.

**Identifying Coordinates of Point A**

We can clearly see that the point A is above – 3 on the x axis. This means that the x coordinate of the point A is – 3. Now, the point A is to the left of 3 on the y axis. This means that the y coordinate of the point A is 3.

**Hence the coordinates of the point A are ( -3, 3 ).**

**Identifying Coordinates of Point B**

We can clearly see that the point B is below – 1 on the x axis. This means that the x coordinate of the point B is – 1. Now, the point B is to the left of – 3 on the y axis. This means that the y coordinate of the point B is – 3.

**Hence the coordinates of the point B are ( – 1, – 3 ).**

**Identifying Coordinates of Point C**

We can clearly see that the point C is above 2 on the x axis. This means that the x coordinate of the point C is 2. Now, the point C is to the right of 4 on the y axis. This means that the y coordinate of the point C is 4.

**Hence the coordinates of the point C are ( 2 , 4 ).**

**Identifying Coordinates of Point D**

We can clearly see that the point D is below 4 on the x axis. This means that the x coordinate of the point C is 4. Now, the point D is to the right of – 4 on the y axis. This means that the y coordinate of the point D is – 4.

**Hence the coordinates of the point C are ( 4 , – 4 ).**

Now let us see how to plot the coordinates of y in accordance with the coordinates of x on the x –y axis.

**Plotting Y Coordinates**

It is important to note here that the order in which you write x- and y-coordinates in an ordered pair is very important. The x-coordinate always comes first, followed by the y-coordinate. Therefore, the ordered pairs ( 3 , 4 ) and ( 4 , 3 ) are two different points.

Hence, we can say that the first number in the brackets relates to the x axis. Also, the second number in the brackets relates to the y axis.

Let us consider an example.

Suppose we want to plot the point ( 2 , 3 ) on the x-y plane.

First let us consider the first point, i..e 2. We know that the first number in the brackets relates to the x axis, therefore, we will highlight 2 on the x-axis as shown below.

Next, we will check the value of the second coordinate. We have been given the value 3. Now, we know that the second number in the brackets relates to the y axis. Therefore, we will mark 3 on the y axis as shown below –

Now, the intersection of these two points on the plane will be out point ( 2 , 3 ) which is shown in the graph below –

**Cartesian Coordinates of a Point**

Let X ‘ O X and Y ‘ O Y be the coordinate axis and let P be any point in the plane of the paper. Let us draw a line P M perpendicular to the X ‘ O X and P N perpendicular to the Y ‘ O Y.

The length of the line segment O M is called the x-coordinate or abscissa of point P and the length of the directed line segment On is called the y-coordinate or ordinate. Let O M =x and O N = y. Then the position of the point P in the plane with respect to the coordinate axis is represented by the ordered pair ( x , y ). This ordered pair ( x , y ) is called the coordinates of point P.

**The Y Intercept**

Before understanding what we mean by the Y intercept let us recall that the intercepts of a graph are points at which the graph crosses the axis. Therefore, by this definition, the *y-***intercept** is the point at which the graph crosses the *y-*axis. At this point, the *x-*coordinate is zero. . The coordinates of the y intercept are always in the form of ( 0 , y ).

Similarly, the *x-***intercept** is defined as the point at which the graph crosses the *x-*axis. At this point, the *y-*coordinate is zero. The coordinates of the x intercept are always in the form of ( x , 0 ).

Let us understand this through an example. Suppose on a graph we wish to plot a line that passes the x axis at -4 units and the y axis at 6 units. In this case, the point on the x axis will be ( -4 , 0 ) and the point on the y-axis will be ( 0, 6 ). The points thus plotted on the graph will be as represented in the figure below –

**How to find a Y Intercept?**

It is important to learn here that there are more than one ways to find the y-intercept. It depends upon the kind of information that is available to us, whether, it is a straight line, a parabola, an ellipse or any other equation. However, in all cases, we solve algebraically to find the y-intercept. Also, since the y-intercept always has a corresponding x-value of 0, replace x with 0 in the equation and solve for y.

**Y Intercept of a Straight Line**

To understand the y intercept of a straight line, let us first recall the equation of a straight line. We know that a straight line is a curve such that every point on the line segment joining any two points on it lies on it. Also, the trigonometrical tangent of the angle that a line makes with the positive direction of the x-axis in an anticlockwise sense is called the slope or the gradient of a line. We also know that,** **if (x_{1}, y_{1}) and (x_{2}, y_{2}) are coordinates of any two points on a line then its slope is given by

m = $\frac{y_2- y_1}{x_2- x_1} = \frac{Difference\:in\:ordinates}{Difference\:in\:adbscissae}$

Let ax + by + c = 0 be a first degree equation in x, y where a, b, c are constants. Let P (x_{1}, y_{1}) and Q(x_{2}, y_{2}) be any point on the curve represented by

a x+ b y + c = 0 ……………………………. ( 1 )

Therefore, if we wish to find out the y intercept through the above general equation of a line , we can do so by substituting x = 0 in the equation ( 1 ), we will get

a ( 0 ) + b y + c = 0

⇒ b y + c = 0

⇒ b y = – c

⇒ y = – $\frac{c}{b}$

**Hence, the y intercept of the equation of a line in general form is given by y = **– $\frac{c}{b}$

**Solved Examples**

**Example 1** Name the ordered pair containing the x and the y coordinates of the points marked in the given graph.

**Solution** We have been given four points, A, B, C and D. We need to find the x coordinate and the y coordinates of these four points. Let us find them one by one.

**Identifying Coordinates of Point A**

We can clearly see that the point A is above 5 on the x axis. This means that the x coordinate of the point A is 5. Now, the point A is to the right of 3 on the y axis. This means that the y coordinate of the point A is 3.

**Hence the coordinates of the point A are ( 5 , 3 ).**

**Identifying Coordinates of Point B**

We can clearly see that the point B is above – 5 on the x axis. This means that the x coordinate of the point B is – 5. Now, the point B is to the left of 4 on the y axis. This means that the y coordinate of the point B is 4.

**Hence the coordinates of the point B are ( – 5 , 4 ).**

**Identifying Coordinates of Point C**

We can clearly see that the point C is below – 5 on the x axis. This means that the x coordinate of the point C is – 5. Now, the point C is to the left of – 5 on the y axis. This means that the y coordinate of the point C is – 5.

**Hence the coordinates of the point B are ( – 5 , – 5 ).**

**Identifying Coordinates of Point D**

We can clearly see that the point D is below 3 on the x axis. This means that the x coordinate of the point D is 3. Now, the point D is to the right of – 2 on the y axis. This means that the y coordinate of the point D is – 2.

**Hence the coordinates of the point B are ( 3 , – 2 ).**

**Example 2** Find the y-intercept of the equation y = 4x^{2} – 3x

**Solution** We have been given the equation, y = 4x^{2} – 3x. we are required to find the y intercept of this equation. We know that in order to find the y intercept we must put x = 0. Therefore, substituting x = 0 in the given equation we have,

y = 4 ( 0 )^{2} – 3 ( 0 )

⇒ y = 0

**Hence, the y intercept of the equation y = 4x**^{2}** – 3x = ( 0 , 0 ) which means that the given equation will pass through the origin.**

**Example 3**** **Plot the coordinate ( 1, -2 ) on the x-y plane.

**Solution**** **We have been given the coordinate ( 1 , – 2 ). We need to plot it on the xy-plane. We will follow the following steps to plot the given coordinate of the xy plane.

- Remember the first number in the brackets relates to the x axis.
- The -1 means move one place, but because there is a minus sign in front of it, this time we move one place to the left, along the x axis.
- The second number in the brackets relates to the y axis.
- The 2 means move two places up, along the y axis.

**Key Facts and Summary**

- A Cartesian coordinate plane is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric elements.
- Let X ‘ O X and Y ‘ O Y be two mutually perpendicular lines through a point O in the plane of a graph paper. The line X O X ‘ is called the x-axis or axis of x and the line Y O Y ‘ is called the y-axis or the axis of y. The two lines X O X ‘ and Y O Y ‘ taken together are called the coordinate axis or the axis of the coordinate system. The point O is called the point of origin.
- The x-axis and the y-axis together divide the entire coordinate system into four equal parts which are called the quadrants of the coordinate system.
- The
*y-***intercept**is the point at which the graph crosses the*y-*axis. At this point, the*x-*coordinate is zero. . The coordinates of the y intercept are always in the form of ( 0 , y ). Similarly, the*x-***intercept**is defined as the point at which the graph crosses the*x-*axis. At this point, the*y-*coordinate is zero. The coordinates of the x intercept are always in the form of ( x , 0 ). - While plotting cooridnates on an xy plane, the x-coordinate always comes first, followed by the y-coordinate.

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