|
|
|
|
|
|
|
|
|
|
|
|
Identify the conic section $$$4 x^{2} - 16 x - 10 y^{2} - 50 y = \frac{133}{2}$$$
Related calculators: Parabola Calculator, Circle Calculator, Ellipse Calculator, Hyperbola Calculator
Your Input
Identify and find the properties of the conic section $$$4 x^{2} - 16 x - 10 y^{2} - 50 y = \frac{133}{2}$$$.
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, $$$A = 4$$$, $$$B = 0$$$, $$$C = -10$$$, $$$D = -16$$$, $$$E = -50$$$, $$$F = - \frac{133}{2}$$$.
The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = 3200$$$.
Next, $$$B^{2} - 4 A C = 160$$$.
Since $$$B^{2} - 4 A C \gt 0$$$, the equation represents a hyperbola.
To find its properties, use the hyperbola calculator.
Answer
$$$4 x^{2} - 16 x - 10 y^{2} - 50 y = \frac{133}{2}$$$A represents a hyperbola.
General form: $$$4 x^{2} - 16 x - 10 y^{2} - 50 y - \frac{133}{2} = 0$$$A.
Graph: see the graphing calculator.