# Which curve is this? Analytical geometry

## ${x}^{2} - 3 \cdot x \cdot y + {y}^{2} - 4 \cdot x - 6 \cdot y + 1 = 0$ Determine the curve this equation is for and the elements of the curve.

Jan 18, 2017

The equation:

${x}^{2} - 3 x y + {y}^{2} - 4 x - 6 y + 1 = 0$

describes an hyperbole

#### Explanation:

The equation:

${x}^{2} - 3 x y + {y}^{2} - 4 x - 6 y + 1 = 0$

is an equation of second degree in $x$ and $y$ and as such, it represents a conic section.

We can determine which section by analyzing the cubic and quadratic invariant:

${I}_{3} = \left\mid \begin{matrix}1 & - \frac{3}{2} & - 2 \\ - \frac{3}{2} & 1 & - 3 \\ - 2 & - 3 & 1\end{matrix} \right\mid = 1 \cdot \left(- 8\right) + \frac{3}{2} \cdot \left(- \frac{15}{2}\right) - 2 \cdot \left(\frac{13}{2}\right) = - 8 - \frac{45}{4} - 13 = - \frac{129}{4} \ne 0$

The cubic invariant is non null, so the curve is non degenerate.

${I}_{2} = \left\mid \begin{matrix}1 & - \frac{3}{2} \\ - \frac{3}{2} & 1\end{matrix} \right\mid = - \frac{5}{4}$

The quadratic invariant is negative, so the curve is an hyperbole.

graph{x^2-3xy+y^2-4x-6y+1=0 [-82.2, 77.8, -43.56, 36.44]}