Quadratic Representation of Pair of Straight Lines/Examples/x^2 - x y - 2 y^2 + 2 x + 5 y = 3

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Examples of Quadratic Representation of Pair of Straight Lines

The equation:

$x^2 - x y - 2 y^2 + 2 x + 5 y = 3$

represents the two straight lines embedded in the Cartesian plane:

\(\ds x + y - 1\) \(=\) \(\ds 0\)
\(\ds x - 2 y + 3\) \(=\) \(\ds 0\)


Proof

\(\ds \paren {x + y - 1} \paren {x - 2 y + 3}\) \(=\) \(\ds 0\)
\(\ds \leadstoandfrom \ \ \) \(\ds x^2 - x \cdot 2 y + 3 x + x y - 2 y^2 + 3 y - x + 2 y - 3\) \(=\) \(\ds 0\) multiplying out
\(\ds \leadstoandfrom \ \ \) \(\ds x^2 - x y - 2 y^2 + 2 x + 5 y\) \(=\) \(\ds 3\) simplifying

The result follows from Quadratic Representation of Pair of Straight Lines.

$\blacksquare$


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