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Standard Equation of Ellipse

The set of re...

Question

The set of real values of $k$ for which the equation $(k + 1) x^{2} + 2 (k - 1) x y + y^{2} - x + 2 y + 3 = 0$ represents an ellipse is

(0, 3)

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(- \infty, 0)

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(3, + \infty)

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(- \infty, + \infty)

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Solution

The correct option is B $(0, 3)$
Given conic is $(k + 1) x^{2} + 2 (k - 1) x y + y^{2} - x + 2 y + 3 = 0$
Here $a = (k + 1), b = 1, c = 3, f = 1, g = - \frac{1}{2}, h = k - 1$
$h^{2} = (k - 1)^{2}, a b = k + 1$
For given conic to be ellipse we must have
$h^{2} < a b$
$(k - 1)^{2} < k + 1 \Rightarrow k^{2} - 3 k < 0 \Rightarrow k (k - 3) < 0$
$\Rightarrow k \in (0, 3)$

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