Show that the equation $x^{2} + 2 x y + 2 y^{2} + 2 x + 2 y + 1 = 0$ does not represent a pair of lines.

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Solution

Comparing the equation
$x^{2} + 2 x y + 2 y^{2} + 2 x + 2 y + 1 = 0$ with
$a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ , we get,
$a = 1, h = 1, b = 2, g = 1, f = 1, c = 1$ .
The given equation represents a pair of lines, if
$D = ∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣ = 0$ and $h^{2} - a b \geq 0$
Now, $D = ∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 1 & 2 & 1 1 & 1 & 1 \end{matrix} ∣ ∣ ∣ ∣$
$= 1 (2 - 1) - 1 (1 - 1) + 1 (1 - 2)$
$= 1 - 0 - 1 = 0$
and $h^{2} - a b = (1)^{2} - 1 (2) = - 1 < 0$
$∴$ given equation does not represent a pair of lines.

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