The equation $8 x^{2} + 12 y^{2} - 4 x + 4 y - 1 = 0$ represents

an ellipse

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a hyperbola

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a parabola

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a circle

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Solution

The correct option is A an ellipse
Given equation is $8 x^{2} + 12 y^{2} - 4 x + 4 y - 1 = 0$
On comparing with
$a x^{2} + b y^{2} + 2 h x y + 2 g x + 2 f y + c = 0$ , we get
$a = 8, b = 12, h = 0, g = - 2, f = 2, c = - 1$
$∴ a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2}$
$= 8 \times 12 \times - 1 + 2 \times 2 \times - 2 \times 0 - 8 {(2)}^{2} - 12 {(- 2)}^{2} + 1 {(0)}^{2}$
$= - 96 - 0 - 32 - 48 + 0 = - 176 \neq 0$
and $h^{2} - a b = 0 - 96 < 0$
Hence, it represents an ellipse.

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The equation $8 x^{2} + 12 y^{2} - 4 x + 4 y - 1 = 0$ represents