Which of the following second degree equation represents a pair of straight lines

x^{2} - x y - y^{2} = 1

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- x^{2} + x y - y^{2} = 1

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4 x^{2} - 4 x y + y^{2} = 4

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x^{2} + y^{2} = 4

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Solution

The correct option is B $4 x^{2} - 4 x y + y^{2} = 4$
For equ: $a x^{2} + b y^{2} + 2 h x y + 2 g x + 2 f y + c = 0$ ,
it represents straight line if
$∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣ = 0$
Check above determinant for options. For $4 x^{2} - 4 x y + y^{2} - 4 = 0$
$a = 4, b = 1, c = - 4, h = - 2, g = 0, f = 0$
$∣ ∣ ∣ ∣ \begin{matrix} 4 & - 2 & 0 - 2 & 1 & 0 0 & 0 & - 4 \end{matrix} ∣ ∣ ∣ ∣$
$∴$ This equation represents straqight line
so, option $C$ is correct.

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