SOLUTION: Rotate the axes and eliminate the xy term of x^2 + 8xy + y^2 - 15 = 0. please include necessary detailed explanations if possible. thank you

Algebra ->  Trigonometry-basics -> SOLUTION: Rotate the axes and eliminate the xy term of x^2 + 8xy + y^2 - 15 = 0. please include necessary detailed explanations if possible. thank you      Log On


   

Question 860401: Rotate the axes and eliminate the xy term of x^2 + 8xy + y^2 - 15 = 0. please include necessary detailed explanations if possible. thank you
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Although most textbooks use x' and y', for convenience I
will use capital X instead of x' and capital Y instead of y'.

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 XY

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

Your equation is

x² + 8xy + y² - 15 = 0.

In this case A-1, B=8, C=1, D=0, E=0, F = -15

Since A=C=1 the required angle of rotation is theta=pi%2F4

The substitutions to make are 

x=X%2Acos%28theta%29-Y%2Asin%28theta%29
y=X%2Asin%28theta%29%2BY%2Acos%28theta%29

Since cos%28pi%2F4%29=sin%28pi%2F4%29=sqrt%282%29%2F2%29, the substitutions are

x=X%28sqrt%282%29%2F2%29-Y%28sqrt%282%29%2F2%29
y=X%28sqrt%282%29%2F2%29%2BY%28sqrt%282%29%2F2%29

or

x=expr%28sqrt%282%29%2F2%29%28X-Y%29
y=expr%28sqrt%282%29%2F2%29%28X%2BY%29

or





x² + 8xy + y² - 15 = 0 becomes



Multiply through by 2

%28X%5E2-2XY%2BY%5E2%29%2B8%28X%5E2-Y%5E2%29%2B%28X%5E2%2B2XY%2BY%5E2%29-30=0

X%5E2-2XY%2BY%5E2%2B8X%5E2-8Y%5E2%2BX%5E2%2B2XY%2BY%5E2-30=0

10X%5E2-6Y%5E2-30=0

10X%5E2-6Y%5E2=30

Divide through by 30 to get 1 on the right side:

X%5E2%2F3-Y%5E2%2F5=1



Edwin