The equation $2 x^{2} - 3 x y + 5 y^{2} + 6 x - 3 y + 5 = 0$ represents

a parabola

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an ellipse

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a hyperbola

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a pair of straight lines

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Solution

The correct option is A an ellipse
Given conic is $2 x^{2} - 3 x y + 5 y^{2} + 6 x - 3 y + 5 = 0$
Here $a = 2, b = 5, c = 5, f = - 3 / 2, g = 3, h = - 3$
$∴ Δ = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 2.5.5 + 2. (- 3 / 2) .3 . (- 3) - 2 (3 / 2)^{2} - 5 (3)^{2} - 5 (- 3)^{2} = - 35 / 2 \neq 0$
and $h^{2} = 9 a b = 10$
clearly $h^{2} < a b$
Therefore, the given conic represents an ellipse.
Hence, option 'B' is correct.

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