How to derive a formula?

Question

Mathematics

How to derive from the formula
Ax² + 2Bxy + Cy² + Dx + Ey + F = 0
this formula: сtg 2a = (AC) / 2B?
(a - angle of rotation of the coordinate axes)

Answer 1

First, by replacing x=x'+x0, y=y'+y0, we can remove the coefficients D and E and assume that the center lies at the point {0, 0}.
Next, you need to choose such a rotation angle that the coefficient at xy becomes zero. Then something like A'x^2+C'y^2 + F' = 0 remains - which means the ellipse is aligned along the axes and this was the desired angle.
Substituting the formulas for the rotation of the coordinate system x=x' cosa-y' sina and y=x' sina + y' cosa and bringing the terms, you can make an equation for this very coefficient in front of xy. It will only depend on A,B,C and sina, cosa. Further, this trigonometric equation must be combed and solved. You will have to use this formula for the cotangent of a double angle :.
You will have an equation with cosines, sines. It can be elementary reduced to tangent. Then apply this formula and get the answer.