Question

A)Calculate the center and radius of the circle by completing the square of the equation x2+y2−8x−6y=−21.(1 point)
Responses

center=(4,3), radius=4
center= left parenthesis 4 comma 3 right parenthesis , radius=4

center=(−4,−3), radius=4
center= left parenthesis negative 4 comma negative 3 right parenthesis , radius=4

center=(8,6), radius=4
center= left parenthesis 8 comma 6 right parenthesis , radius=4

center=(−8,−6), radius=4

Answer 1

To find the center and radius of the circle given by the equation \(x^2 + y^2 - 8x - 6y = -21\), we can complete the square for both \(x\) and \(y\).

### Step 1: Rearrange the equation
First, move \(-21\) to the other side:
\[
x^2 - 8x + y^2 - 6y = -21
\]

### Step 2: Complete the square for \(x\) terms
For \(x^2 - 8x\):
- Take half of \(-8\) (which is \(-4\)), square it to get \(16\).
- Therefore, we rewrite \(x^2 - 8x\) as:
\[
(x - 4)^2 - 16
\]

### Step 3: Complete the square for \(y\) terms
For \(y^2 - 6y\):
- Take half of \(-6\) (which is \(-3\)), square it to get \(9\).
- Therefore, we rewrite \(y^2 - 6y\) as:
\[
(y - 3)^2 - 9
\]

### Step 4: Substitute back into the equation
Now substitute back into the equation:
\[
(x - 4)^2 - 16 + (y - 3)^2 - 9 = -21
\]

Combine the constants:
\[
(x - 4)^2 + (y - 3)^2 - 25 = -21
\]
\[
(x - 4)^2 + (y - 3)^2 = 4
\]

### Step 5: Identify the center and radius
From the equation \((x - h)^2 + (y - k)^2 = r^2\), we can identify:
- Center: \((h, k) = (4, 3)\)
- Radius: \(r = \sqrt{4} = 2\)

### Conclusion
The center of the circle is \((4, 3)\) and the radius is \(2\).

None of the provided multiple-choice responses match the calculated answer directly, but based on the calculation, you have:

**Center: \( (4, 3) \), Radius: \( 2 \)**