# Identify the conic with th e given equa­tion and give its equation in standard form 6x^2 - 4xy + 9y^2 - 20x - 10y - 5 = 0

Identify the conic with th e given equa­tion and give its equation in standard form $$6x^2 - 4xy + 9y^2 - 20x - 10y - 5 = 0$$

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odgovoreh
Given $$6x^2 - 4xy + 9y^2 - 20x - 10y - 5 = 0$$ All degenerate conic sections have equations of the form $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$.
$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ where A,B,C,D,E and F are constants. If $$B^2 − 4AC$$ is less than zero, if a conic exists and A = C, it will be a circle If $$B^2 − 4AC$$ is less than zero, if a conic exists , it will be ellipse. If $$B^2 − 4AC$$ equals zero, if a conic exists, it will be a parabola. If $$B^2 − 4AC$$ is greater than zero, if a conic exists, it will be a hyperbola. Here in equation $$6x^2 - 4xy + 9y^2 - 20x -10y - 5 = 0$$, $$B^2 - 4 AC = 4^2 - 4 \cdot 6 \cdot 9$$
$$= 16 - 216$$
$$= - 200 < 0$$ and $$A \neq C$$ Hence it is an ellipse