Question

determine the center and radius of the following circle and sketch the graph. x^2 + y^2 = - 6x - 8y

Explanation:

Step1: Rewrite the equation

Rearrange the given equation $x^{2}+y^{2}=-6x - 8y$ to the standard - form of the circle equation $(x - a)^{2}+(y - b)^{2}=r^{2}$. First, move all terms to one side: $x^{2}+6x+y^{2}+8y = 0$.

Step2: Complete the square for x - terms

For the $x$ - terms in $x^{2}+6x$, we use the formula $(x + m)^{2}=x^{2}+2mx+m^{2}$. Here, $2m = 6$, so $m = 3$ and $m^{2}=9$. Add 9 to both sides of the equation: $x^{2}+6x + 9+y^{2}+8y=9$.

Step3: Complete the square for y - terms

For the $y$ - terms in $y^{2}+8y$, since $2n = 8$, then $n = 4$ and $n^{2}=16$. Add 16 to both sides of the equation: $x^{2}+6x + 9+y^{2}+8y+16=9 + 16$.

Step4: Write in standard form

The left - hand side can be factored as $(x + 3)^{2}+(y + 4)^{2}=25$.

Answer:

The center of the circle is $(-3,-4)$ and the radius $r = 5$.