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.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $13$. Quadratic equation representing a pair of straight lines