# Use the Discriminant Test to determine the type of the conic section (in every case non degenerate equation). Plot the curve if you have a computer algebra system.displaystyle{2}{x}{2}-{8}{x}{y}+{3}{y}{2}-{4}={0}

Use the Discriminant Test to determine the type of the conic section (in every case non degenerate equation). Plot the curve if you have a computer algebra system.
$2{x}^{2}-8xy+3{y}^{2}-4=0$

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Step 1
Given that the equation is $2{x}^{2}-8xy+3{y}^{2}-4=0$
It is known that the general equation of the conic
$A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ and the discriminant is given by
$\mathrm{\Delta }={B}^{2}-4AC$
If $\mathrm{\Delta }\ne 0$ then the equation is non degenerate.
${B}^{2}-4AC>0,$ equation represent hyperbola
${B}^{2}-4AC=0$, equation represent parabola
${B}^{2}-4AC<0$, equation represent ellipse
Step 2
It's required to find the type of the conic. Here $A=2,B=-8,C=3.$
$\mathrm{\Delta }={B}^{2}-4AC$
$={\left(-8\right)}^{2}-4\left(2\right)\left(3\right)$
$=64-24$
$=40$
$>0$
Thus, the given equation is hyperbola.
Step 3
Sketch the graph of the equation $2{x}^{2}-8xy+3{y}^{2}-4=0$.