The equation [tex]3x^2 + xy - 9y^2 + 2x + 5 = 0[/tex] is a hyperbola.
We have given that,
[tex]4x^2 + 3x + 4y^2+ 5 = 0 \\2x^2 - 5xy + 4y^2 + 10x - y + 12 = 0 \\3x^2 - 6xy + 3y^2 + 5x -2y + 1 = 0 \\3x^2 + xy - 9y^2 + 2x + 5 =0[/tex]
What is hyperbola?
A hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows.
The equation [tex]3x^2 + xy - 9y^2 + 2x + 5 = 0[/tex] is a hyperbola.
Therefore option D is correct.
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