Chapter 10.5, Problem 13AYU

To determine

The graph of each equation without completing their squares: $6x²+3y²-12x+6y=0$.

Answer to Problem 13AYU

Ellipse.

Explanation of Solution

Given:

$6x²+3y²-12x+6y=0$

Formula used:

$Ax²+Cy²+Dx+Ey+F=0$

where $A$ and $C$ are not both equal to zero:

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

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

c. Defines a hyperbola if $AC<0$.

Calculation:

Consider the given equation $6x²+3y²-12x+6y=0$.

$A=6\text{ and }C=3$; $AC=\left(6\right)\left(3\right)=18>0$

Since $AC>0$ and $A\ne C$, the equation defines an ellipse.