Different Forms of the Equation of a Line in Coordinate Geometry

In this very article, we are going to discuss various forms of the equation of a line. A coordinate plane consists of an infinite number of points. If we consider a point P(x,y) in a 2d plane and a line named it as N. Then what we will determine is that the point we consider lies on the line L or it lies above or below of the line. That’s when straight-line comes into this scenario. Here we will include the important topic related to the equation of a line in different forms. 

Forms of the Equation of the Line

Based on the parameters known for the straight line, there are 5  forms of the equation of a line that is used to determine and represent a line's equation:

• Point Slope Form –

This form requires a point on the line and the slope of the line. The referred point on the line is (x1,y1) and the slope of the line is (m). The point is a numeric value and represents the x coordinate and the y coordinate of the point and the slope of the line (m) is the inclination of a line with the positive x-axis.

Here, (m) can have a positive, negative, or zero slope. Hence, the equation of a line is as follows:

( y - y11 ) = m( x - x11)

• Two Point Form –

This form is a further explanation of the point-slo

on of a line passing through the two points - (x11, y11), and (x22, y22) is in this way:

(y−y1)=(y2−y1)(x2−x1)(x−x1)(y−y1)=(y2−y1)(x2−x1)(x−x1)

• Slope Intercept Form –

The slope-intercept form of the line is y = mx + c. And here, 'm' is the slope of the line and 'c' is the y-intercept of a line. This line cuts the y-axis at the point (0, c), where c is the distance of this point on the y-axis from the origin.

The slope-intercept form is an important form and has great applications in the different topics of mathematics.

y = mx + c

• Intercept Form –

The equation of a line in this form is formed with the x-intercept (a) and the y-intercept (b). The line cuts the x-axis at a point (a, 0), and the y-axis at a point(0, b), and a, b are the respective distances of these points from the origin. While these two points can be substituted in a two-point form and simplified to get this intercept form of the equation of a line.

The intercept form of the equation of the line explains the distance at which the line cuts the x-axis and the y-axis from the origin.

• Normal Form –

The normal form is based on the line perpendicular to the given line, which passes through the origin, is known as the normal.

Here, the parameters of length of the normal is 'p' and the angle made by this normal is 'θ' with the positive x-axis is useful to form the equation of a line. The normal form of the equation of the line is in this way:

xcosθ + ysinθ = P

Different Forms of the Equation of a Straight Line

A. Equation of Line Parallel to the y-axis

Equation of a straight line which is parallel to the y-axis at a distance of ‘a’ then the equation of y-axis will be x=a (here ‘a’ is a coordinate in the plane).

Consider this  example Equation of line parallel to y-axis for coordinate (7,8) is x=8

 B. Equation of Line Parallel to the x-axis

Equation of a straight line if the straight line is parallel to the x-axis the equation will be y=a where ‘a’ is an arbitrary constant.

To understand one can consider this example, consider this a point (9,10) Equation of line parallel to the x-axis is x=9

 C. Point- slope Form of an Equation

Let a line passing through a particular point Q(X1, Y1) and P(X, Y) be any point present in the mentioned line.

The slope of a line= Y - Y1/X – X2

And by the definition m is the slope,

Hence, m = Y - Y1/X – X2

On comparing Y – Y1 = m(X – X1) is the required point-slope form equation of a line

 D. Equation of the Line in Two-point Form

Consider an arbitrary constant P(x,y) present in the line L and the Line L passes through two points A(x1,y1) and B(x2,y2). We consider ‘m’ as the slope of the line L.

m= y2-y1 / x2- x1

Then the equation of the line is

y2-y1 = m(x2-x1)

Substituting the value of m we get

y-y1={ y2- y1/ x2-x1}(x-x1)

Equation of the required line in two point form is y - y1= y2- y1/ x2 - x1(x -x1).

E. Equation of a Line in Intercept Form

Let AB line cuts intercept on the x-axis at (a, 0) and on the y-axis at (0, b)

From two-point form:

ð  y = -b/a (x – a)

ð  y = b/a ( a – x)

ð  x/ a + y/b = 1 is the required equation of line in intercept form

Example:

Consider finding the equation of a line which has made an intercept of 4 in x axis and has made a cut of y-axis in the graph

Solution

So, b = -3  and a = 4

ð  x/4 + y/-3 = 1

ð  3x – 4y = 12 hence the required equation of a line in intercept form

Slope-intercepts Form of a Line:

Consider a line L whose slope be m which cuts an intercept on the y-axis at the distance of ‘a’. hence the point is (0, a)

Hence, the required equation is:

ð  y – a = m(x – 0)

ð  y = mx + a which is the required equation of a line.

Example:

Find the equation of a line which has a  slope of -1 and has an intercept of 4 units in the positive section of the y-axis.

Solution

Here, m = -1 and a = -4

Substituting this value in y = mx + a we get:

ð  y = -x – 4

ð  x + y + 4 = 0

Solved Examples

Example

Determine the equation of a line which passes through the point (-4, -3) and it is parallel to the x-axis.

Solution

Here, m = 0, X1 = -4, Y1 = -3.

Through the above equation: Y + 3 = 0(X + 4)

ð  Y = -3 is the required equation

Example

Find the equation of the line joining by the points  (4,-2) and  (-1,3).

Solution: here the two given points are (X1,Y1) = (-1,3) and (X2,Y2)= (4,-2)

Equation of line in two point form is

ð  y – 3 = { 3 – (- 2)/ -1 – 4 }( x+1)

ð   - x – 1 = y – 3

ð  x + y – 2 = 0.