SOLUTION: Find to the nearest degree, the acute angle of rotation 1 that eliminates the xy term of -3x^2 + 3xy + 5y^2 - x + 2y - 6 = 0

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Question 860396: Find to the nearest degree, the acute angle of rotation 1 that eliminates the xy term of -3x^2 + 3xy + 5y^2 - x + 2y - 6 = 0
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Find to the nearest degree, the acute angle of rotation 1 that eliminates the xy term of -3x^2 + 3xy + 5y^2 - x + 2y - 6 = 0
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Form:: Ax^2 +Bxy + Cy^2 + Dx + Ey + F = 0
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Angle of rotation:: cot(2t) = (A-C)/B = (-3-5)/3 = -8/3
2t = arc(tan(-3/8)) = -20.56 = 339.44
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Ans: t = 169.72
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Cheers,
Stan H.

Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor gave you an obtuse angle, not an acute angle.

The angle of rotation theta of the graph 

Ax² + Bxy + Cy² + Dx + Ey + F = 0

necessary to transform it into an equation in x' and y' which contains
no term in x'y'

is calculated by tan%282theta%29=B%2F%28A-C%29 or pi%2F4 if A = C

Your equation is

-3x² + 3xy + 5y² - x + 2y - 6 = 0

In this case A=-3, B=3, C=5, D=-1, E=2, F = -6

tan%282theta%29=B%2F%28A-C%29

tan%282theta%29=3%2F%28-3-5%29

tan%282theta%29=3%2F%28-8%29

tan%282theta%29=-3%2F8

Find the reference angle for 2theta,
using the inverse tangent function of a
calculator using abs%28-3%2F8%29=3%2F8. That gives
reference angle for 2theta as 20.55604522

So the smallest positive value for 2theta
that has tangent -3%2F8 is in QII, and is
180-20.55604522 = 159.4439548

So

2theta=%22159.4439548%B0%22

and

theta=%2279.72197739%B0%22

To nearest degree, 80°

Edwin