Identify the conic section $4 x^{2} + 9 y^{2} = 36$

Identify and find the properties of the conic section $4 x^{2} + 9 y^{2} = 36$.

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, $A = 4$, $B = 0$, $C = 9$, $D = 0$, $E = 0$, $F = -36$.

The discriminant of the conic section is $\Delta = 4 A C F - A E^{2} - B^{2} F + B D E - C D^{2} = -5184$.

Next, $B^{2} - 4 A C = -144$.

Since $B^{2} - 4 A C \lt 0$, the equation represents an ellipse.

To find its properties, use the ellipse calculator.

$4 x^{2} + 9 y^{2} = 36$A represents an ellipse.

General form: $4 x^{2} + 9 y^{2} - 36 = 0$A.

Graph: see the graphing calculator.