The conic represented by the equation $13 x^{2} - 18 x y + 37 y^{2} + 2 x + 14 y - 2 = 0$ is

a circle

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an ellipse

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a parabola

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a hyperbola

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Solution

The correct option is B an ellipse
On comparing the given equation with the equation $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
we get,
$a = 13, b = 37, c = - 2, f = 7, g = 1, and h = - 9$
$△ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$
$= - 962 - 126 - 637 - 37 + 162 \neq 0$
and $h^{2} - a b = 81 - 13 \times 37 < 0$
Thus, we have
$△ \neq 0 and h^{2} < a b .$
So, the given equation represents an ellipse.

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