Conic Section Calculator

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

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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.