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Line in a Coordinate Geometry

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In the coordinate Geometry two lines are present one is known as vertical line and other is defined as horizontal line. Now, we will see what is vertical and horizontal line in a coordinate geometry? If the x- coordinate of a line remains unchanged and y –coordinate changes according to the coordinate value then that line is known as vertical line. A line which goes straight up , down and also parallel to the y – axis of the coordinate plane is said to be vertical line. All points lie on the line having same x – coordinate. In the coordinate geometry no Slope is defined for vertical lines.
Let’s see the equation of a vertical line which is given by:
Equation of a line is x = u;
Where, ‘x’ is the coordinates of any Point on the line and ‘u’ is the line which crosses x – axis.
A line whose y- coordinates remains unchanged and x – coordinate changes according to the coordinate value is known as horizontal line.
A line which goes straight left and right and which is also parallel to the x – axis of the Coordinate Plane is said to be horizontal line. All points lie on the line having same y – coordinate. In the coordinate geometry, slope is defined for horizontal line and the value of Slope of horizontal line is zero.
The equation of a horizontal line is given by:
Equation of a line is y = v;
Where, ‘y’ is the coordinates of any point on the line and ‘v’ is the line which crosses the x – axis. In the mathematics, the coordinate geometry is fully dependent on horizontal and vertical lines. This is all about the line in a coordinate geometry.

How to Define a Line?

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In Geometry, a line is one dimensional figure made up of a Point extend along a fixed direction and the opposite to the same direction without end. It has no width. The shortest distance between any two points on a plane is called the Straight Line. It has no width. To understand how to define a line we use the following figure of a line.


In the above figure, the line PQ passes through the points P and Q, is perfectly straight and goes off continuously in both directions forever.
When we draw a line on the plane or say page, we show it as a line with an arrow head on each end as shown in the above figure above. These arrow heads shows that the line goes off to infinity in both directions. A Line Segment has two end points and it has a fixed length. It is a part of a line which has two end points. On the other hand a Ray has one end point and extends along to infinity in one direction.

In another branch of mathematics called coordinate geometry, line is defined by the coordinates. These are the two Numbers, x-coordinate and y-coordinate that show where the points are located. The two lines are present in a Coordinate Plane graph which is vertical and horizontal line. The line whose x- coordinate remains same and y –coordinate changes is known as vertical line. Equation of a vertical line is x = p. It goes straight up and down and also parallel to the y – axis.

A line whose y- coordinate remains same and x – coordinate changes is known as horizontal line. A line which move straight left and right and also parallel to the x – axis of the coordinate plane is known as horizontal line. The Slope of a horizontal line is zero.
The equation of a horizontal line is given by, y = q.
Where, ‘y’ denotes the coordinates of any point on the line and ‘q’ denotes the line which crosses the x – axis.

How to Define a Line Using Two Points

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Line is a one dimensional object with negligible width. Line is bounded by two end points one is starting Point and other is ending point. Let us see how to define a line using two points.
Assume that we have two pints of a line; starting points and ending points. Starting points are (x1, y1) and ending points are (x2, y2). Using these two pints we can define a line. Equation of a line is:
y = mx + c.

This is general equation of a line. We can define the line equation by other form also, we will discuss that later. If we have two points of a line then we can represent the line as:
Y = y1 + [(y2 – y1) / (x2 – x1)] . (x2 – x1),
Here (y2 – y1) / (x2 – x1) is the Slope of line. In this equation points x1 and x2 are assumed as different points, just in case the points are equal then we will assume that x = x1 and second point will not be necessary any more.

We can also write this equation as:
(y – y1)= [(y2 – y1) / (x2 – x1)] . (x2 – x1),
In simpler form:
(y – y1) * (x2 – x1) = (y2 – y1) . (x2 – x1),
Now we can convert this equation to simplest form which is very easy to remember, when we have two points of line (x1, y1) and (x2, y2) now equation can be written as:
(y – y1) / (y2 – y1) = (x – x1) / (x2 – x1).

Using One Point and a Slope

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Point - slope form is a method of representing a line in form of equation. We can plot points on graph by plugging in values of x, point - slope form makes whole process easier. Point - slope form can also be taken to make a graph and find the equation of that characteristics line.

Point Slope form uses a single Point on graph and slope of line.
Standard point - slope formula is shown below:

y – y1 = m (x – x1), here value of variable ‘m’ denotes slope of line, variable y1 does not Mean that variable 'y' is multiplied by 1. It means 1 is subscript of y.
We will understand it with the help of an example:

For example: Suppose we have points (5, 4) and slope of line is 3, then calculate the equation of line in point slope form:
Solution: Given points (5, 4) and Slope of a line is 3. As we discussed above that formula to find the standard point slope is given as:

y – y1 = m (x – x1),
Here value of x1 is 5 and value of y1 is 4. Now put these values in above given formula:
y – y1 = m (x – x1),
After solving result look will be:
y – 4 = 3 (x – 5),
This is the process of Using one point and a slope.