Conic Section Calculator
Solve conic sections step by step
The calculator will identify the given conic section (non-degenerate or degenerate) and find its discriminant, with steps shown. Also, it will graph the conic section.
Related calculators: Parabola Calculator, Circle Calculator, Ellipse Calculator, Hyperbola Calculator
Your Input
Identify and find the properties of the conic section $$$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$$$.
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, $$$A = 7$$$, $$$B = -2$$$, $$$C = 7$$$, $$$D = -22$$$, $$$E = -38$$$, $$$F = 67$$$.
The discriminant of the conic section is $$$\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = -2304$$$.
Next, $$$B^{2} - 4 A C = -192$$$.
Since $$$B^{2} - 4 A C \lt 0$$$, the equation represents an ellipse.
To find its properties, use the ellipse calculator.
Answer
$$$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$$$A represents an ellipse.
General form: $$$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$$$A.
Graph: see the graphing calculator.