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Identify the conic section $$$x^{2} - \frac{\left(y - 7\right)^{2}}{8} = 1$$$
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Your Input
Identify and find the properties of the conic section $$$x^{2} - \frac{\left(y - 7\right)^{2}}{8} = 1$$$.
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, $$$A = 1$$$, $$$B = 0$$$, $$$C = - \frac{1}{8}$$$, $$$D = 0$$$, $$$E = \frac{7}{4}$$$, $$$F = - \frac{57}{8}$$$.
The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = \frac{1}{2}$$$.
Next, $$$B^{2} - 4 A C = \frac{1}{2}$$$.
Since $$$B^{2} - 4 A C \gt 0$$$, the equation represents a hyperbola.
To find its properties, use the hyperbola calculator.
Answer
$$$x^{2} - \frac{\left(y - 7\right)^{2}}{8} = 1$$$A represents a hyperbola.
General form: $$$x^{2} - \frac{y^{2}}{8} + \frac{7 y}{4} - \frac{57}{8} = 0$$$A.
Graph: see the graphing calculator.