I have a formula any and wonder what that is equation (hyperbola, point, lines, ellipse, parabola etc.) . However, I have doubts when I do the translation and rotation of coordinate systems.
I know it's parallel lines but what to do to keep my resolution? I don't know what I do from this moment
$$ x² + 4y² + 4xy - 1 = 0 \\ \therefore A = 1, B = 4, C = 4, D = 0, E = 0, F = -1 $$
$$ B^2 - 4AC = 0 \\ 16 - 4.1.4 = 0 $$ So it's line, parallel lines, parabola or empty
$$ A \neq C, tan\theta = \dfrac{B}{A - C} \rightarrow \theta \cong 33,6º\\ X = ucos\theta - vsin\theta\\ Y = usen\theta + vcos\theta\\ (0,83u - 0,55v)² + 4(0,55u + 0,83v)² + 4(0,83u - 0,55v)(0,55u + 0,83v) - 1 = 0 $$