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}

Question

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.
$2 x^{2} - 8 x y + 3 y^{2} - 4 = 0$

Answer 1

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

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.displaystyle{2}{x}{2}-{8}{x}{y}+{3}{y}{2}-{4}={0}

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