Analytic Geometry

[kakamilan17]

I'm trying to find the angle of rotation of this equation. x²-3xy+4y²+7=0
I was given x=x'cosθ - y'sinθ and y=x'sinθ+y'cosθ


So I plug in those two equations for x and y and get a really long equation.
I simplified, then I factored out x'y' out of the long equation and was told to set the coefficient of x'y' to 0. So I end with:

cos²θ + 2cosθsinθ - sin²θ = 0

Now I'm stuck trying to use the trig identities to solve for the angle.
I could solve this using logic, but as this problem is for Analytic Geometry, I'm supposed to prove everything using math. So how can I solve for the angle in this case? I can't seem to be able to make headway with the identities.

So which identities could I use to solve this last part?

-Thank you very much

 

[soroban]

Hello, kakamilan17!

I'm trying to find the angle of rotation of this equation: .\(\displaystyle x^2-3xy+4y^2+7\:=\:0 \)

You should have been given this information . . .


\(\displaystyle \text{Given: }\:Ax^2 + Bxy + Cy^2 + Dx + Ey + F \:=\:0\)

\(\displaystyle \text{The angle of rotation }\theta\text{ is given by: }\:\tan2\theta \:=\:\dfrac{B}{A-C}\)

\(\displaystyle \text{The discriminant is: }\:\Delta \:=\:B^2 - 4AC\)

. . \(\displaystyle \Delta = 0:\:\text{parabola}\)
. . \(\displaystyle \Delta < 0:\:\text{ellipse}\)
. . \(\displaystyle \Delta > 0:\:\text{hyperbola}\)