Question 723467
The discriminant of the equation {{{Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0}}} is the quantity {{{B^2 - 4AC}}}.

Assuming {{{A}}}, {{{B}}} and {{{C}}} are {{{not}}}{{{ all}}} {{{ 0}}} and that degenerate cases are excluded.

Then,

if {{{B^2 - 4AC < 0}}}, the graph is an {{{ellipse}}} or a {{{circle}}}
if {{{B^2 - 4AC = 0}}}, the graph is a {{{parabola}}}
if {{{B^2 - 4AC > 0}}}, the graph is a {{{hyperbola}}}

 you have 

{{{6x^2-2y^2+24x+2y-1=0}}}.....where {{{A=6}}}, {{{B=0}}},{{{C=-2}}},{{{D=24}}},{{{E=2}}}, and {{{F=-1}}}

{{{B^2 - 4AC=0^2 - 4*6*(-2)=-24*(-2)=48}}}

so, {{{B^2 - 4AC > 0}}}, the graph is a {{{hyperbola}}}

{{{ graph( 600, 600, -10, 10, -10, 10, 6x^2-2y^2+24x+2y-1>=0) }}}