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Which curve is this? Analytical geometry

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

Answer 1

Samantha Adkison

The equation:

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

describes an hyperbole

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]}

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Answer 2

Aurora Alvarado

The curve described is a parabola.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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.

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