Chapter 10.5, Problem 19AYU

To determine

The graph of each equation without completing their squares: $x²+y²-8x+4y=0$.

Answer to Problem 19AYU

Circle.

Explanation of Solution

Given:

$x²+y²-8x+4y=0$

Formula used:

$Ax²+Cy²+Dx+Ey+F=0$

where $\text{A}$ and $\text{C}$ are not both equal to zero:

a. Defines a parabola if $\text{A}\text{C}=0$.

b. Defines an ellipse (or a circle) if $\text{A}\text{C}>0$.

c. Defines a hyperbola if $\text{A}\text{C}<0$.

Calculation:

Consider the given equation $x²+y²-8x+4y=0$.

$A=1\text{ and }C=1$; $AC=\left(1\right)\left(1\right)=1>0$

Since $AC>0$ and $\text{A}=\text{C}$, the equation defines a circle.