Question 1136333
recall:

If {{{b^2-4ac<0}}}, then the equation represents an {{{ellipse}}}.

-A subordinate special case of this occurs when {{{A=C }}}and{{{ B=0}}}, then the equation represents a {{{circle}}}.

- If {{{b^2-4ac=0}}}, then the equation represents a{{{ parabola}}}.

- If {{{b^2-4ac >0}}}, then the equation represents a {{{hyperbola}}}.

-A subordinate special case of this occurs, when {{{A+C=0}}}, then the equation represents a rectangular hyperbola.


you are given:

{{{4x^2 - 9xy - 5y^2 + 15 = 0 }}}...rearrange  the terms
 {{{ - 5y^2- 9xy+ 4x^2+ 15 = 0 }}}

{{{a=-5}}}, 
{{{b=-9x}}}, 
{{{c=4x^2+15}}}

discriminant is:

{{{b^2-4ac=(-9x)^2-4*(-5)(4x^2+15)}}}

{{{b^2-4ac=81x^2+20(4x^2+15)}}}

{{{b^2-4ac=81x^2+80x^2+300}}}

{{{b^2-4ac=161x^2+300}}}

=> {{{b^2-4ac > 0}}} and your equation represents a {{{hyperbola}}}


{{{ graph( 600, 600, -10, 10, -10, 10, -(1/10)(-sqrt(161x^2+300)-9x),(1/10) (-sqrt(161x^2+300)-9x)) }}}