4. Write an equation of the circle in standard form and determine the center and radius of

the circle. Show all work.

𝑥 2 + 𝑦 2 − 10𝑥 + 6𝑦 − 15 = 0

Center: _______________

Radius:________________

We are given the equation of the circle:

\[
x^2 + y^2 - 10x + 6y - 15 = 0
\]

### Step 1: Group \(x\) and \(y\) terms together
\[
(x^2 - 10x) + (y^2 + 6y) = 15
\]

We want to rewrite this equation by completing the square for both \(x\) and \(y\).

### Step 2: Completing the square

**For the \(x\) terms**:
\[
x^2 - 10x
\]
To complete the square, take half of \(-10\), which is \(-5\), then square it to get \(25\).

So, add \(25\) to both sides of the equation.

**For the \(y\) terms**:
\[
y^2 + 6y
\]
Take half of \(6\), which is \(3\), then square it to get \(9\).

So, add \(9\) to both sides of the equation.

The equation now looks like this:
\[
(x^2 - 10x + 25) + (y^2 + 6y + 9) = 15 + 25 + 9
\]
\[
(x - 5)^2 + (y + 3)^2 = 49
\]

### Step 3: Write the equation in standard form
The standard form of a circle is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
Here, \(h = 5\), \(k = -3\), and \(r^2 = 49\), so \(r = 7\).

### Final Answer:
- **Center**: \( (5, -3) \)
- **Radius**: \( 7 \)

Would you like more details or have any questions?

Here are some related questions for practice:
1. What is the equation of a circle with center \((-4, 2)\) and radius 6?
2. How do you complete the square on a quadratic expression?
3. How do you find the distance between two points on the coordinate plane?
4. If a circle has a radius of 10, what is its area?
5. How does the general form of a circle's equation differ from its standard form?

**Tip**: Completing the square is essential for converting quadratic equations into forms that reveal geometric properties like the center and radius of a circle!

Write an equation of the circle in standard form and determine the center and radius of the circle. Show all work. x^2 + y^2 - 10x + 6y - 15 = 0 Center: _______________ Radius: ________________

Learn how to find the center and radius of the circle by converting the general form equation x^2 + y^2 - 10x + 6y - 15 = 0 into standard form. This step-by-step guide involves completing the square and applying geometric properties. Suitable for Grades 9-10 students.

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