Question

In Problem, identify the graph of each equation without completing the squares.

$6x^2+3y^2-12x+6y=0$

Short answer

The equation represents graph of an ellipse.

Step-by-step solution

Step 1. Given information

An equation is$6x^2+3y^2-12x+6y=0$.

Step 2. Identify the graph.

The equation $A{x}^2+C{y}^2+Dx+Ey+F=0$where $A$and $C$both cannot be zero,

(a) Defines a parabola if $AC=0$.

(b) Defines an ellipse (or a circle) if $AC>0$.

(c) Defines a hyperbola if $AC<0$

Comparing the given equation with the equation $A{x}^2+C{y}^2+Dx+Ey+F=0$we can see that $A=6$and $C=3$So, we get $AC=18$.

Since $AC>0$, the given equation is the equation of an ellipse.