Rearrange into . Now it's in the form you can solve for y in terms of x
13x^2+ 6√3xy + 7y^2 – 16 = 0
I'm sorry if this isn't algebra, I assumed it was. If it is Calculus or something else, please direct me to the right place.
I'm looking for the answer, but I'm looking for how to do it more than I am searching for the answer. So please be thorough in your response.
Since you put this in the Linear Algebra section I am assuming you know what a quadratic form is. This can be rewritten in matrix form as
Now the coefficient matrix is real symmetric so the matrix is orthogonally diagonalizable.
Now diagonalize the coefficient matrix I got the two eigenvalues
Find the two linearly independent eigenvectors and normalize them to have length 1.
This will give you the orthogonal matrix
Now back to the equation.
written in matrix form gives
Remember that for an orthogonal matrix its transpose is its inverse now make the substitution
subbing this into the above equation gives
Which now gives a diagonal matrix in our new coordinate system we have
OR
which is an ellipse
This gives
This can now be plotted in the rotated coordinate system.