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# Coordinate of a Point

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 Sub Topics A number pair which represents the location of a Point in two - dimensional space is known as coordinates of a point. Suppose we have coordinate of a point R (5, -9) then it defines the location of points. Where ‘R’ is the name and the Numbers in the bracket is the ‘x’ and ‘y’ coordinates. The first number or x – coordinate denotes how far the horizontal axis is and y- coordinates denote how long up and down the axis is. The x – axis or horizontal is also known as Abscissa of the points. The y – axis or vertical horizontal is also known as ‘ordinate’ of the points. When we have two pairs in a plane and we want to find the Distance between two points then we can find the distance between two points. Formula is given for finding the distance between two points is: D = √ (u2 - u1)2 + (v2 - v1)2; And in the case of three coordinates (u1, v1, w1) and (u2, v2, w2) the formula for finding the distance is given by: D = √ (u2 - u1)2 + (v2 - v1)2+ (w2 - w1)2; Now we will see how to find the distance between the points; Let the coordinates of points are (-3, -2) and (1, 6) then we have to find the distance between two points. We know that the coordinates of the points is (u1, v1) and (u2, v2); Here the value of u1 = -3; And the value of u2 = 1; The value of v1 = -2; The value of v2 = 6; Then the distance between two points is: The formula for finding the distance between two points is: D = √ (u2 - u1)2 + (v2 - v1)2; Then put the values in the given formula: D = √ (1 - 5)2 + (6 – (-2))2; On further solving we get: D = √ (-4)2 + (8)2; D = √ 16 + 64; D = √80; D = 8.94 So the distance is 8.94. Using this formula we can find the distance between points.

## Abscissa

Before understanding the meaning of abscissa coordinate it is necessary to know about the coordinate plane. A rectangular plane which consists of two number lines i.e. vertical number line and the horizontal number line is known as Coordinate Plane. As we know that horizontal line is represented by the x - axis which is also said to be abscissa and vertical line is known as Ordinate which is the y - axis. In plane the number line comprises of Positive and Negative Numbers. In case of horizontal line positive numbers are placed at right hand side and negative number are placed at left hand side. While in the vertical number line upper side contains positive numbers and lower side contains negative numbers. The abscissa meaning is the distance from the y -axis in the Cartesian coordinate system which is measured parallel to the x - axis.
A Point in a plane which has coordinates (4, 3), so ‘4’ is abscissa of this coordinate plane.
Abscissa definition can be understood by definition given below;
As we know that two axes and four quadrants are present in a rectangular plane also sometimes it represents the element of Ordered Pair, suppose (7, 2) here first element which is ‘7’ is considered as the abscissa and it is plotted on the horizontal axis in two dimensional coordinate system. The second element which is 2 in the ordered pair is considered as ‘ordinate’ which is plotted on vertical line i.e. on y - axis.
In the middle of 16th century the word ‘abscissa’ was first used since then it became the standardized mathematical term. Without using abscissa or ordinate it is not possible to plot the points on the graph.
Suppose we have plotted the point (2, -4), this point lies in fourth quadrant. The point 2 represents the ‘abscissa’ and -4 represents the ‘ordinate’. The point is located in fourth quadrant. As we know that in fourth quadrant abscissa is positive and ordinate is negative. So we can say abscissa meaning is x -coordinate.
This is all about abscissa definition.

## Ordered Pair

A Set of Numbers that is used to initiate a Point on a coordinate plane is called as ordered pair. In other words, we can say that two numbers written in the proper definite order, generally the number written in the parentheses in this form (5, 8), then numbers is known as ordered pair.
An ordered pair Math is basically written in the form of (a, b) where 'a’ denotes the x – coordinate and ‘b’ represents the y - coordinate.
The value of ‘x’ coordinate is always along to the horizontal axis and the value of ‘y’ is along to the vertical axis in the coordinate Geometry.
For example: (6, -3), (12, -2), (18, -13) are the example of order pair.
Suppose (a1, b1) and (a2, b2) be the order pair, then the characteristics properties of the order pair is given by:
⇒ (a1, b1) = (a2, b2); if and only if a1 = a1 and b1 = b1.
In the Coordinate Plane the set of order pair in which first entry lie in the set ‘x’ and second entry lie in the set ‘y’ then it can be Cartesian product of ‘x’ and ‘y’ and in between the set ‘x’ and set ‘y’ a binary relation is a subset of X x Y.
Now we will see the ‘Wiener’s definition of ordered pairs:
Norbert Wiener invented first set theoretical definition of the order pair in the year 1994 and the definition of order pair says:
⇒ (a, b) : = a, ?, b .
Now we will see the Hausdorff’s definition of order pair. In the year 1994 Hausdorff invented the definition of order pair as:
⇒ (a, b) : = a, 1, b, 2; where the numbers 1 and 2 are the two distinct objects which are obtained from ‘a’ and ‘b’.
This is all different types of definition of order pair.

Coordinate axis 'x' and 'y'. y- axis is generally perpendicular to x- axis . Point of Intersection is called origin or center with coordinate (0, 0). Plane of coordinate system is divided into four quadrants like 1st quadrant where x- axis and y- axis both are positive, that means if we draw any Point in 1st quadrant then we get coordinate points with positive x and y- coordinate points. In second quadrant we have x- axis as negative and y- axis as positive, points will be like (-x, y). Similarly coordinate point in third quadrant will have x- axis as negative and also negative y- axis. So, coordinate point will be like (-x, -y). Now point in fourth quadrant will have positive x- axis and negative y- axis. Point in fourth quadrant is like: (x, -y). Let us look at The Coordinate plane Quadrants with all the four quadrants marked in it. It is shown below:

In above figure there are four points. (2, 2) in first quadrant where both are positive points, (-2, 1) in second quadrant where 'x' is negative and 'y' is positive as discussed, (-1, -3) in third quadrant where both points are negative points as we have discussed earlier and fourth point (3,-2) in fourth quadrant where 'x' is 3 and it is positive and 'y' is 2 which is negative.

## Coordinate of a Point on a Circle

Circle is a figure that represents a conic section. The general equation of a Circle can be given as: (X - H)2 + (Y - K)2 = R2. Where, (H, K) represents the Center of Circle other than origin and 'R' is the Radius of Circle. If center of the circle was (0, 0), equation can be written as: (X)2 + (Y)2 = R2. A circle can be defined as Set of all those points which are equidistant from center and they are constant, equal to radius. A circle being a 2 – dimensional figure can be plotted in Rectangular Coordinate System. So, any Point lying on circle can also be plotted if we know its coordinates. For instance, coordinate of a point on a circle are (x, y). A point lying on circle means it lies on circumference of circle. So two conditions are fulfilled by point:
1. The point will satisfy the equation of circle.
2. Its distance from center of circle will be equal to the radius of circle and can be calculated using distance formula.
Let us consider an example of a point (x, y) at an angle 600 lying on a circle with equation:
(X - 3) 2 + (Y - 4) 2 = 25,
Here, center of the circle is: (3, 4).
And radius of circle is: R = 5.
According to these two conditions that were satisfied by point we can write:
(x - 3) 2 + (y - 4) 2 = 25 equation 1 and
Adding x - coordinate of center of the circle to the product 5 cos 60 we get: 3 + 5 cos 60.
Similarly, adding y - coordinate of center of circle to the product 5 sin 60 we get: 4 + 5 sin 60.
So coordinates of point lying on the circle at an angle 600 is given as (3 + 5 cos 60, 4 + 5 sin 60).

## Coordinate of a Point on a Line

Coordinate of a Point on a line is calculated with the help of section formula or Ratio formula.Let us consider a point 'P' which divides a Line Segment between two points 'A' and 'B' in a ratio 'a' and 'b'. A point can cut the line externally or internally.
Section formula for same is defined below:

Ratio formula when Line Segment is divided externally:
(aP2 – bP1 / a - b), (aQ2 – bQ1 / a – b),
Ratio formula when line segment is divided internally:
(aP2 + bP1 / a + b), (aQ2 + bQ1 / a + b),
Here in above two formulas: 'a' and 'b' are ratio of line segment and (P1, Q1) and (P2, Q2) are coordinates of end points.
Now let us consider a small example in which line segment is divided by a point externally, ratio is 2: 1 and coordinates of points of two end points of line segment is (3, 4) and (2, 3). Now we have to find coordinates of that point. Now using above formula, we can say that:
(P1, Q1) are (3, 4) and point (P2, Q2) are (2, 3) and ratio 'a' and 'b' are 1 and 2. Now putting values in above formula:
(aP2 – bP1 / a - b), (aQ2 – bQ1 / a – b),
We get: (1 * 2 – 2 * 3/ 1 – 2), (1 * 3 – 2 * 4 / 1 – 2),
Now solving them further we get the coordinates as (4, 5).
So, we can say that coordinates of a point dividing the line segment externally is (4, 5).
Similarly we can calculate the Coordinate of a Point when point divides the line segment internally.
So on putting above values in formula for point dividing the line internally, we get:
(aP2 + bP1 / a + b), (aQ2 + bQ1 / a + b),
We get: (1 * 2 + 2 * 3 / 1 + 2), (1 * 3 + 2 * 4 / 1 + 2). So we get coordinates as: (8/3, 11/3).