Normal Form of a Line

Introduction:

The Normal form of a Line, is a line perpendicular to the tangent line at a point where function intersects with the respective tangent line.

Let’s consider m as the slope of the tangent line. The normal form of a line is the line that is exactly perpendicular to its tangent line. And the perpendicular must be realised at the point where the tangent line and function intercepts. 

Now, in this case, when m is the tangent line’s slope, then the slope of the normal line will be the negative reciprocal of m. This means the slope of the line is -1/m.

Equations of a line are written in the following standard forms:

• Slope-intercept form
• Intercept form
• Normal form

We now look at all the straight line formulas and all the different forms of finding the straight-line equation in detail.

Normal Form of a Line calculator

The equation of the straight line upon which the length of the perpendicular from the origin is p and this perpendicular makes an angle α with the x-axis is x cos α + y sin α = p

If the line length of the perpendicular drawn from the origin upon a line and the angle that the perpendicular makes with the positive direction of x-axis be given then we can easily find the equation of a line:x cos α + y sin α = p

The general form of the line is : Ax+By+C=0

so,

A cos α = B sin α = – p

As a result, cos α = -p/A and sin α = -p/B.

Furthermore, we can infer that,

cos^2α + sin^2α = (p/A)^2 + (p/B)^2

1 = p^2 ((A^2 + B^2  )/A^2 . B^2)

p=ABA^2 +B^2

We can conclude the following from the general equation of a straight line Ax + By + C = 0:

• The slope is given by -A/B, given that B ≠ 0
• The x-intercept is given by -C/A and the y-intercept is given by -C/B
• If two points (x1, y1) and(x2,y2) lie on the same side of the line Ax + By + C = 0, then the expressions (Ax1+ By1 + C). ( Ax2 + By2 + C)>0 , else these points would lie on the opposite sides of the line Ax + By + C = 0

General Equation of a Line

Two variables of the first degree can be used to derive a line in the general equation.

Ax + By +C = 0,

A, B ≠ 0 where A, B, and C are constants belonging to real numbers.

A straight line is always the result of representing the equation geometrically.

The following is an illustration of straight-line formulas in different forms:

Straight Line Formulas

The following is a summary of the straight line formulas we have discussed so far:

Slope (m) of a non-vertical line passing through the points (x1, y1 ) and (x2, y2 )	m=(y2-y1)/(x2-x1), x1≠x2
Equation of a horizontal line	y = a or y=-a
Equation of a vertical line	x=b or x=-b
Equation of the line passing through the points (x1 ,y1 ) and (x2, y2 )	y-y1= [(y2-y1)/(x2-x1)]×(x-x1)
Equation of line with slope m and intercept c	y = mx+c
Equation of line with slope m makes x-intercept d.	y = m (x – d).
Intercept form of the equation of a line	(x/a)+(y/b)=1
The normal form of a line formula	x cos α+y sin α = p

Straight Line: Example

The following examples will help you better understand this concept:

(1). Determine the slope and both intercepts of the equation of a line 2x – 6y + 3 = 0.

Solution:

Using slope-intercept form, the given equation 2x – 6y + 3 = 0 can be expressed as:

y = x/3 + 1/2

On comparing it with y = mx + c,

Then the slope of the line, m = 1/3

Furthermore, we can reframe the above equation in intercept form as follows:

x/a + y/b = 1

2x – 6y = -3

x/(-3/2) – y/(-1/2) = 1

Thus, x-intercept = -3/2 and y-intercept = 1/2.

(2) An equation of a line is represented by 13x – 13y + 12 = 0. Compute the slope and both intercepts.

Solution: The given equation 13x –13 y + 12 = 0 can be represented in slope-intercept form as:

y = (13x + 12)/13

On comparing it with y = mx + c,

Then the slope of the line, m = 1

Also, the above equation can be reframed in intercept form as:

x/a + y/b = 1

13x –13 y = -12

x/(-12/13) + 13y/(12/13) = 0

Thus, x-intercept = -12/13 and y-intercept  = 12/13.

Conclusion

Lines have a general form of y = mx + c, where m is the gradient, and c is the value where the line crosses the y-axis. C is commonly referred to as the intercept on the y-axis. That is the important point. An equation for a straight line with gradient m and intercept c is y = mx + c on the y-axis. What purpose does it serve? It is used principally to indicate the gradient of a line. In other words, it is used to define a particular line compactly.