# 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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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.

$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$A represents an ellipse.
General form: $7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$A.