Rotating coordinate axes

[Alix]

The instructions are: Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle. x² - 3y = 4, φ = 60°.


I started the problem by using the formulas x = X cos φ - Y sin φ and y = X sin φ + Y cos φ. Substituting the angles, this gave me x = X(1/2) - y(sqrt 3/2) and y = X(sqrt 3/2) + Y(1/2).


I know the next step involves substitution, but I can't figure out what I need to do. The answer to this problem is X² + sqrt 3 XY + 2. Any help or tips as to how to solve this would be greatly appreciated.

 

[Yogi]

The instructions are: Determine the equation of the given conic...
The answer to this problem is X² + sqrt 3 XY + 2.

Did you type everything correctly, there is no equation in your answer. Is it supposed to be X² + sqrt 3 XY + 2=0? or X² + sqrt 3 XY = 2? Please double check everything. Also I know when you type sqrt 3 XY that you probably mean (sqrt 3)XY and not sqrt(3XY) based on what type of problem it is but please include grouping symbols anyways.

 

[Alix]

I checked again and you're correct, it is X² + (√3)XY + 2 = 0.

 

[Yogi]

You are right, substitute the values for x and y that you got into the problem then simplify and combine like terms, I don't think that answer is right though.

x² - 3y = 4
[X(1/2) - Y(sqrt 3/2)][X(1/2) - Y(sqrt 3/2)] - 3[X(sqrt 3/2) + Y(1/2)] = 4
Multiply it out combine like terms etc...

I don't think the given answer is correct though.

 

[Alix]

Thanks for your help, that really cleared it up for me. I tried the same method with another problem and it worked fine, so I think the answer may have been mistyped.