The general form of a conic section is:
Ax²+Bxy+Cy²+Dx+Ey+F=0
If B does not equal 0, then the conic section has been rotated. In this problem, we have
x² + xy + y² = 10
Thus: B = 1 (coefficient of xy), A = 1 (coefficient of x²), and C = 1 (coefficient of y²).
And the angle of rotation, θ, can be found using the formula:
cot (2θ) = (A - C)/B
Notice that "cot (2θ)" represents the reverse of the tangent function, that is, 1/tan(2θ).
Now, let's use A = B = C = 1 in that formula:
cot (2θ) = (1 - 1)/1 = 0/1 = 0
Since we obtained zero, that means the tangent of 2θ is infinity. And the angle for which this is true is 90º ( π/2).
Then,
2θ = 90º
θ = 90º/2
θ = 45º
