Question

Which curve is this? Analytical geometry

`x^2-3*x*y+y^2-4*x-6*y+1=0`
Determine the curve this equation is for and the elements of the curve.

Answer 1

The equation:

`x^2-3xy+y^2-4x-6y+1=0`

describes an hyperbole

Explanation:

The equation:

`x^2-3xy+y^2-4x-6y+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 = abs((1, -3/2, -2), (-3/2, 1, -3), (-2, -3, 1)) = 1* (-8)+3/2*(-15/2)-2*(13/2) = -8-45/4-13 =-129/4 !=0`

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

`I_2 = abs((1, -3/2), (-3/2, 1)) = -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]}