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PRE-CALCULUS
LESSON 7-6
NORMAL FORM OF A
LINEAR EQUATION
Pg. 463-469
Objectives:
1) Write the standard form of a linear equation given the length of the normal and the angle it makes
with the x-axis
2) Write linear equations in normal form.
KEY CONCEPT:
1. Normal Line: A line that is perpendicular to another line, curve, or surface and passes
through the origin. (This is the shortest distance from the line to the origin.)
2. Normal Form: The equation of a line written in terms of length of the normal, depicted
from the initial line to the origin (0, 0) (Using trigonometric properties to derive)
This integrates your past knowledge of Algebra I with lessons learned in Pre-Calc. This diagram
should illustrate the “normal line” and its relationship with any other line.
1. CO is the “normal line” segment
2.  (phi) the positive angle rotated from
the x-axis to the normal line
3. “p” is the length of the “normal line”
4. MC is the segment from the intersection
of the “normal line” and the given line
down perpendicular to the x-axis
Therefore use trigonometry to determine the various elements required to write the equation of
either the normal form or standard form of the original line (given the length of the normal and
the positive rotated angle of the normal)
1. cos  = OM  p cos  = OM
p
2. sin  = MC  p sin  = MC
p
3. The “slope” of OC is Δy = MC = p sin  = sin
Δx
OM
p cos 
cos
4. “Slope” of the original line is “perpendicular” to OC so the “negative reciprocal”  = – cos 
sin 
* Remember the “Point Slope” Equation from Algebra I?
y – y1 = m(x – x1) where in our illustration (x1, y1) is Point “C” or (p cos , p sin ) and m is
the negative reciprocal
Using Substitution of (x1, y1) and m, we can produce the “standard form of a linear equation” of
the original line
1) y – y1 = m(x – x1)
2) y – psin = – cos  (x – p cos )
sin 
Multiply both sides by (sin )
3) y sin  – psin2  = – x cos  + p cos2 
Rearrange
4) x cos  + y sin  = psin2  + p cos2 
Factor out “p” and remember that sin2x+ cos2x= 1
5) x cos  + y sin  = p (1)
6) x cos  + y sin  – p = 0 This is the original line in Standard Form
*** We can now write the Standard Form of a linear equation if given the values of  and p
(Look at Example 1 on page 464 to illustrate this principle.)
Now try page 467 Problem 18: Given p = 5 and  = 4π
3
x cos 4π + y sin 4π – 5 = 0
3
3
_
x (-1/2) + y(-√3/2) – 5 = 0
Multiply both sides times -2 to clear fractions
_
x + √3 +10 = 0
This is displayed in the form: Ax + By + C = 0
We can then change this Standard Form into Normal Form by simply dividing by each term by
_______
+ √A2 + B2 (**The sign selected here will be the opposite of that of the “C” term)
(Page 465 describes this process)
(Look at Example 3 on page 466 to illustrate how to do this process)
_
Now try page 467 Problem 26: Given: - √3 x – y = 2
Move our “C” term to left side
__
_______
________
__
2
2
2
2
- √3 x – y – 2 = 0
+ √A + B = √(√3) + 1 = √4 = 2
_
- √3 x – y – 2 = 0
2
2 2
_
 - √3 x – y – 1 = 0
2
2
Now find the length of the normal and the angle it makes with the positive x-axis
Remember this from above? x cos  + y sin  – p = 0
Match it to our new data and determine the needed information…
The length of the normal is “p” so p = 1 The angle is the cos and sin  so  = 210 (from our
chart data) The signs of  will aid in identifying the correct angle to select.