CBSE Grade 10 Maths Chapter 7 - Coordinate Geometry.


What is Cartesian System ?

When two perpendicular lines i.e. one horizontal line and one vertical line intersect each other at their zeroes , they form a Cartesian Plane. These two perpendicular lines are called the coordinate axis.

  • The horizontal line is known as the x-axis

  • The vertical line is known as the y-axis.

  • The point where these two lines intersects each other is called the center or the origin of the coordinate plane. 

  • Origin is represented as ‘O’ and it's coordinates are (0, 0).

Coordinates of a Point

To write the coordinates of a point we need to follow these rules-

  • The x - coordinate of a point is marked by drawing perpendicular from the y-axis measured a length of the x-axis .It is also called the Abscissa.

  • The y - coordinate of a point is marked by drawing a perpendicular from the x-axis measured a length of the y-axis .It is also called the Ordinate.

  • While writing the coordinates of a point in the coordinate plane, the x - coordinate comes first, and then the y - coordinate. We write the coordinates in brackets.

 

In the above figure, 
 
OB = CA = x coordinate (Abscissa)
CO = AB = y coordinate (Ordinate)

We write the coordinate as (x, y).

Note : As the origin O has zero distance from the x-axis and the y-axis so its abscissa and ordinate are zero. Hence the coordinate of the origin is (0, 0).

Quadrants of the Cartesian Plane

The Cartesian plane is divided into four quadrants named as Quadrant I, II, III, and IV anticlockwise.

 

 I - ( +,+) means 1st quadrant is enclosed by the positive x-axis and the positive y-axis.
II - (-,+) means 2nd quadrant is enclosed by the negative x-axis and the positive y-axis.
III - (-,-) means 3rd quadrant is enclosed by the negative x-axis and the negative y-axis.
IV - (+, -) means 4th quadrant is enclosed by the positive x-axis and the negative y-axis

Plotting a Point in the Plane if its Coordinates are Given

Steps to plot the point (2, 3) on the Cartesian plane -

  • Firstly, draw the Cartesian plane by drawing the coordinate axes with 1 unit = 1 cm.

  • To mark the x coordinates, starting from 0 moves towards the positive x-axis and count to 2.

  • To mark the y coordinate, starting from 2 moves upwards in the positive direction and count to 3.

  • Now this point is the coordinate (2, 3)

Likewise, we can plot all the other points, like (-3, 1) and (-1.5,-2.5) in the figure shown below.

 Distance Formula

 The distance between two points which are on the same axis (x-axis or y-axis), is given by the difference between their ordinates if they are on the y-axis, else by the difference between their abscissa if they are on the x-axis.
 
To calculate the distance between any two points A(x1,y1) and B(x2,y2) we use formula shown below in the figure.

where D is the distance between any two points A(x1,y1) and B(x2,y2).
 
 Note : 
 
1) A and B are two separate points.
2) The final formula to calculate the distance between two points still remains the same,irrespective of which quadrant point A and B lie in .

Equation of a Straight Line

An equation of line is used to plot the graph of the line on the Cartesian plane.

The equation of a line is written in slope intercept form as

y = mx +b

where m is the slope of the line ( also known as Gradient ) and b is the y intercept.

Distance from Origin

If we have to find the distance of any point from the origin then, one point is P(x,y) and the other point is the origin itself, which is O(0,0).

 Section Formula

If the point P(x, y) divides the line segment joining A(x1,y1) and B(x2,y2) internally in the ratio m:n, then, the coordinates of P are given by the section formula as:
 
Two things can be achieved with the help of a section formula in coordinate geometry. 
  • The ratio in which the point divides the given line segment can be found if we know the coordinates of that point. 
  • Also, it is possible to find the point of division if we know the ratio in which the line segment joining two points has given.

Section formula is used to determine the coordinate of a point that divides a line segment joining two points into two parts such that the ratio of their length is m:n.

 Proof of  Section Formula

 Let P(x1,y1) and Q(x2,y2) be two points in the XY-plane. Let M (x,y) be the point which divides the line segment PQ internally in the ratio m:n.
 
 
 

Ratio in which the point divides the line segment 

Midpoint 

The midpoint of any line segment divides it in the ratio 1:1
 

Points of Trisection

 Centroid of a triangle

 

 Area from Coordinates

Area of a triangle can be found using three different methods. The three different methods are discussed below

Method 1

When the base and altitude of the triangle are given.

Area of the triangle, A = bh/2 square units.

Where b and h are base and altitude of the triangle, respectively.

Method 2


Method 3

 

 

 What are Collinear Points?

Collinear points are the points that lie on the same line. If two or more than two points lie on a line close to or far from each other, then they are said to be collinear, in Euclidean geometry. The points which do not lie on the same line are called non-collinear points.
There are three methods to find the collinear points. They are:
  1. Distance Formula
  2. Slope Intercept Formula
  3. Area of triangle 

Using Distance Formula - 

If A,B and C are three collinear points, then,
Distance from A to B + Distance from B to C = Distance from A to C
AB + BC = AC

 Now, by the Distance Formula we know, to calculate the distance between any two points A(x1,y1) and B(x2,y2) we use formula shown below in the figure.

where D is the distance between any two points A(x1,y1) and B(x2,y2).
 
Hence, we can easily find the distance between the points A ,B and C with the help of this formula.

Using Slope Formula

Three or more points are said to be collinear if the slope of any two pairs of points is the same. The slope of the line basically measures the steepness of the line.

Suppose, X, Y and Z are the three points, with which we can form three sets of pairs, such that, XY, YZ and XZ are three pairs of points. Then, as per the slope formula

If Slope of XY = Slope of YZ = Slope of XZ, then the points X, Y and Z are collinear

Note: Slope of the line segment joining two points say (x1, y1) and (x2, y2) is given by the formula:

m = (y2 – y1)/ (x2 – x1) 

Example - Show that the three points P(2, 4), Q(4, 6) and R(6, 8) are collinear.

Solution: If the three points P(2, 4), Q(4, 6) and R(6, 8) are collinear, then slopes of any two pairs of points, PQ, QR & PR will be equal.

Now, using slope formula we can find the slopes of the respective pairs of points, such that;

Slope of PQ = (6 – 4)/ (4 – 2) = 2/2 = 1

Slope of QR = (8 – 6)/ (6 – 4) = 2/2 = 1

Slope of PR = (8 – 4) /(6 – 2) = 4/4 = 1

As we can see, the slopes of all the pairs of points are equal.

Therefore, the three points P, Q and R are collinear.

Using the Area of Triangle Formula

If the area of a triangle formed by three points is zero, then they are said to be collinear. It means that if three points are collinear, then they cannot form a triangle.

Some Solved Examples

Question : Is the coordinates (x, y) = (y, x)?

Solution :

Let x = (-4) and y = (-2)

So (x, y) = (- 4, – 2)

(y, x) = (- 2, - 4)

Let’s mark these coordinates on the Cartesian plane.

 

You can see that the positions of both the points are different in the Cartesian plane. So,

If x ≠ y, then (x, y) ≠ (y, x), and (x, y) = (y, x), if x = y.

Example:

Plot the points (6, 4), (- 6,- 4), (- 6, 4) and (6,- 4) on the Cartesian plane.

Solution:

As you can see in (6, 4) both the numbers are positive so it will come in the first quadrant.

For x coordinate, we will move towards the right and count to 6.

Then from that point go upward and count to 4.

Mark that point as the coordinate (6, 4).

 

 Similarly, we can plot all the other three points.