Preliminaries in plane geometry (Part 2)

In this post we will discuss about equations of a line in different form. Every equation is related to other equation and can be derived from one other. In the post I have derived the different forms of equations of a straight line. I have started from an equation which is called two-point form and end with the normal form. normal form is also called perpendicular form. I have given graphs of four equations.

The equation of a line can be found if we know anyone of

1. two points
2. one point and slope of the line
3. slope of the line and y intercept
4. angle made by the perpendicular and its length to the line from the origin

The corresponding four types of equations are as follows

1. two points form: (y−y₁)/(x−x₁) = (y₂−y₁)/(x₂−x₁)
2. point-slope form: (y−y₁) = m (x−x₁)
3. slope-intercept form: y = mx + c
4. normal form: x cos α + y sin α = p

The four equations in the graph are
y = 3x + 4, y = x + 7 , y = 2x− 5 and y = −x + 3.

Let A(x₁, y₁) and B(x₂, y₂) be two points. Then the four types of equations can be framed as follows:

1. As the slope of the line will be constant. So, if a variable point is (x,y) then
   (y−y₁)/(x−x₁) = (y₂−y₁)/(x₂−x₁)


2. If the value of the slope is m then we can substitute,
   (y₂−y₁)/(x₂−x₁) = m
   and get the equation of the line as
   (y−y₁)/(x−x₁) = m
   (y−y₁)= m (x−x₁)
   


3. Expanding (y−y₁)=m(x−x₁)
   y−y₁= mx − mx₁
   y = mx−(mx₁− y₁)
   As [−(mx₁ − y₁)] is a constant and can be substituted for c
   y = mx + c
   
   From the above equation we get y = c when x = 0. Hence, c is the y intercept.


4. Normal form is found by considering the angle which the perpendicular from the origin  to the line makes with the x-axis and its length.
   
   
   
   
   In the figure the angle is α and the length of the perpendicular is p. Equation of line in two point form is
   (x₁, y₁) ≡ (a,0) and (x₂, y₂) ≡ (0,b)
   
   Using the two point form
   (y−y₁)/(x−x₁) = (y₂−y₁)/(x₂−x₁)
   y/(x − a) = b/(− a)
   −ay = bx − ab
   bx + ay = ab
   x/a + y/b = 1
   a = p sec α
   b = p cosec α
   x/(p sec α) + y/(p cosec α) = 1
   x cos α + y sin α = p