Identify the conic section $$$164 x^{2} - 216 x y - 16 x + 72 y^{2} + 31 = 0$$$
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Identify and find the properties of the conic section $$$164 x^{2} - 216 x y - 16 x + 72 y^{2} + 31 = 0$$$.
Solution
The general equation of a conic section is $$$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$$$.
In our case, $$$A = 164$$$, $$$B = -216$$$, $$$C = 72$$$, $$$D = -16$$$, $$$E = 0$$$, $$$F = 31$$$.
The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = -576$$$.
Next, $$$B^{2} - 4 A C = -576$$$.
Since $$$B^{2} - 4 A C \lt 0$$$, the equation represents an ellipse.
To find its properties, use the ellipse calculator.
Answer
$$$164 x^{2} - 216 x y - 16 x + 72 y^{2} + 31 = 0$$$A represents an ellipse.
General form: $$$164 x^{2} - 216 x y - 16 x + 72 y^{2} + 31 = 0$$$A.
Graph: see the graphing calculator.