The second degree equation $x^{2} + 3 x y + 2 y^{2} - x - 4 y - 6 = 0$ represent a pair of straight lines

True

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False

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Solution

The correct option is A True
If $⎡ ⎢ ⎣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ⎤ ⎥ ⎦$ = 0 then the given equation represents a pair of straight lines.
Here $⎡ ⎢ ⎣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 3 / 2 & - 1 / 2 3 / 2 & 2 & - 2 - 1 / 2 & - 2 & - 6 \end{matrix} ⎤ ⎥ ⎦$
= 1(-12 - 4) - 3/2(-9 - 1) - 1/2(-3 + 1)
= -16 + 15 + 1 = 0.
$∴$ True.

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