Questions: For the equation x^2 + y^2 - 8x - 6y - 11 = 0, do the following. (a) Find the center (h, k) and radius r of the circle. (b) Graph the circle (c) Find the intercepts, if any. (a) The center is (Type an ordered pair.) The radius is r = (b) Use the graphing tool to graph the circle. (c) Find the intercepts, if any. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The intercept(s) is/are (Type an ordered pair. Use a comma to separate answers as needed. Type exact answers for each coordinate, using radicals as needed.) B. There is no intercept.

Solution Steps

The given equation is \( x^2 + y^2 - 8x - 6y - 11 = 0 \). To find the center and radius of the circle, we need to rewrite this equation in the standard form of a circle's equation, which is \( (x-h)^2 + (y-k)^2 = r^2 \).

To rewrite the equation in standard form, we complete the square for both \(x\) and \(y\).

1. Group the \(x\) and \(y\) terms: \[ x^2 - 8x + y^2 - 6y = 11 \]

2. Complete the square for \(x\): \[ x^2 - 8x \] Add and subtract \((\frac{-8}{2})^2 = 16\): \[ x^2 - 8x + 16 - 16 \]

3. Complete the square for \(y\): \[ y^2 - 6y \] Add and subtract \((\frac{-6}{2})^2 = 9\): \[ y^2 - 6y + 9 - 9 \]

4. Rewrite the equation with the completed squares: \[ (x^2 - 8x + 16) + (y^2 - 6y + 9) = 11 + 16 + 9 \] \[ (x - 4)^2 + (y - 3)^2 = 36 \]

From the standard form equation \((x - h)^2 + (y - k)^2 = r^2\), we can identify the center \((h, k)\) and the radius \(r\).

1. The center \((h, k)\) is \((4, 3)\).
2. The radius \(r\) is \(\sqrt{36} = 6\).

Final Answer