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Answer:
See attached
Step-by-step explanation:
You want the perpendicular form of the given equations, and the corresponding values of p and α.
Beginning with a equation in standard form, we can obtain the perpendicular form as follows.
standard form: ax +by = c
perpendicular form: x·cos(α) +y·sin(α) = p
Ordinarily "standard form" has a, b, c mutually prime and a > 0. For the purpose here, the "standard form" we want has c ≥ 0, so as to facilitate computation of a positive value for p.
The values of p and α are found as ...
[tex]p=\dfrac{c}{\sqrt{a^2+b^2}}\\\\\\\alpha=\tan^{-1}\left(\dfrac{b}{a}\right)\quad\text{with attention to quadrant}[/tex]
In the attached spreadsheet, we have rearranged the given equations to the desired "standard form," and implemented the above formulas for calculating p and α. The roots are expressed as numerical values. The angles (α) are shown in degrees, because the numerical value in radians is not usually a recognizable angle.
Here are a couple of examples of the equations written out in perpendicular form:
(c) x·cos(30°) +y·sin(30°) = 5
(e) x·cos(240°) +y·sin(240°) = 2
