The equation $(x + y + 1)^{2} + k (x^{2} + y^{2} + 1) = 0$ represents two straight lines if the value of $k$ is

0

only

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3

only

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0

- 3

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None of these

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Solution

The correct option is C $0$ or $- 3$
$(x + y + 1)^{2} + k (x^{2} + y^{2} + 1) = 0$
$(k + 1) x^{2} + (k + 1) y^{2} + 2 x y + 2 y + 2 x + (k + 1) = 0$
This equation represents a pair of straight lines then,
Discriminant $= D = a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$
$∴ (k + 1)^{3} + 2 (1 \times 1 \times 1) - (k + 1) - (k + 1) - (k + 1) = 0$
$(k + 1)^{3} - 3 (k + 1) + 2 = 0$
$k^{3} + 3 k^{2} + 3 k + 1 - 3 k - 1 = 0$
$k^{2} (k + 3) = 0$
$k = 0$ or $k = - 3$

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