Question

determine the center and radius of the following circle and sketch the graph. ( x^2 + y^2 = 6x - 8y ) use the graphing tool on the right to graph the equation. click to enlarge graph

Explanation:

Step1: Rearrange the equation

We start with the equation \(x^{2}+y^{2}=6x - 8y\). To complete the square, we move all the \(x\) and \(y\) terms to the left side: \(x^{2}-6x + y^{2}+8y=0\).

Step2: Complete the square for x

For the \(x\) terms, \(x^{2}-6x\), we take half of \(- 6\) which is \(-3\), square it to get \(9\). So we add \(9\) to both sides.

Step3: Complete the square for y

For the \(y\) terms, \(y^{2}+8y\), we take half of \(8\) which is \(4\), square it to get \(16\). So we add \(16\) to both sides.

Step4: Rewrite the equation

After adding \(9\) and \(16\) to both sides, the equation becomes \((x - 3)^{2}+(y + 4)^{2}=9 + 16\). Simplifying the right side, we have \((x - 3)^{2}+(y + 4)^{2}=25\).

The standard form of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.

Answer:

The center of the circle is \((3,-4)\) and the radius is \(5\) (since \(r^{2}=25\), so \(r = 5\)).