Question

HI guys.

find the equations of the lines represented by

4x^2 - 5xy - 6y^2 + 11x + 11y - 3 = 0

I am lost. What two lines are they asking for? And is there a general formula/appraoch to simplifying the duel equation with powers to 2?
I tried turning the x^2 and y^2 into squares but then I was left with the 5xy not knowing what to do. Please help.

Answer 1

It is a conic. First use the discriminant theorem to figure out what conic it is
where
A x^2 + B x y + C y^2 + D x + E y + F = 0
if
B^2 -4 A C
negative - ellipse
0 - parabola
positive - hyperbola
here B^2 - 4 A C = 25 - 4*4*(-6)
= 25 + 96 = +121 so it is a hyperbola
the axis of the hyperbola is at an angle to the x axis though, resulting in the xy term with coefficient -5
They may be asking for the directions of x' and y' that make this a decoupled hyperbola with no xy term. I do not know.