Identify the conic section $$$9 x^{2} - 36 x - 41 y^{2} - 32 y = 124$$$
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Your Input
Identify and find the properties of the conic section $$$9 x^{2} - 36 x - 41 y^{2} - 32 y = 124$$$.
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, $$$A = 9$$$, $$$B = 0$$$, $$$C = -41$$$, $$$D = -36$$$, $$$E = -32$$$, $$$F = -124$$$.
The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = 226944$$$.
Next, $$$B^{2} - 4 A C = 1476$$$.
Since $$$B^{2} - 4 A C \gt 0$$$, the equation represents a hyperbola.
To find its properties, use the hyperbola calculator.
Answer
$$$9 x^{2} - 36 x - 41 y^{2} - 32 y = 124$$$A represents a hyperbola.
General form: $$$9 x^{2} - 36 x - 41 y^{2} - 32 y - 124 = 0$$$A.
Graph: see the graphing calculator.