Problem 11-17: Identify the Center and Radius of the Circle

We are given the equation of the circle:

$x^2 + y^2 - 9x + 4y = -8.25$

Step 1: Rearrange and complete the square

To rewrite this equation in the standard form of a circle, we need to complete the square for both the $x$-terms and the $y$-terms.

Start by grouping the $x$ and $y$ terms:

$(x^2 - 9x) + (y^2 + 4y) = -8.25$

Step 2: Complete the square for $x$-terms

For $x^2 - 9x$, take half of $-9$, square it, and add it to both sides:

$\frac{-9}{2} = -4.5, \quad (-4.5)^2 = 20.25$

So, we add 20.25 to both sides:

$(x^2 - 9x + 20.25) + (y^2 + 4y) = -8.25 + 20.25$

Step 3: Complete the square for $y$-terms

For $y^2 + 4y$, take half of 4, square it, and add it to both sides:

$\frac{4}{2} = 2, \quad (2)^2 = 4$

Add 4 to both sides:

$(x^2 - 9x + 20.25) + (y^2 + 4y + 4) = -8.25 + 20.25 + 4$

Step 4: Simplify

Simplifying both sides gives us:

$(x - 4.5)^2 + (y + 2)^2 = 16$

Step 5: Identify the center and radius

Now the equation is in the standard form of a circle:

$(x - h)^2 + (y - k)^2 = r^2$

Where $(h, k)$ is the center and $r$ is the radius. From the equation $(x - 4.5)^2 + (y + 2)^2 = 16$, we can see that:

• Center: $(4.5, -2)$
• Radius: $r = \sqrt{16} = 4$

Final Answer (boxed):

• Center: $(4.5, -2)$
• Radius: $4$

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Problem 18-20: Graph the Circle with Given Center and Radius

Given:

• Center $C(3, -2)$
• Radius = 5.5 units

Step 1: Equation of the circle

The standard form of the equation of a circle is:

$(x - h)^2 + (y - k)^2 = r^2$

Where $(h, k)$ is the center, and $r$ is the radius. Plugging in the given center $(3, -2)$ and radius $5.5$:

$(x - 3)^2 + (y + 2)^2 = (5.5)^2 = 30.25$

So the equation of the circle is:

$(x - 3)^2 + (y + 2)^2 = 30.25$

Step 2: Graphing the circle

To graph the circle:

1. Plot the center at $C(3, -2)$.
2. Mark points that are 5.5 units away from the center in all directions (left, right, up, down).
3. Draw the circle passing through those points.

If you'd like a graphical representation, I can create an accurate plot for you. Would you like me to proceed with that?

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Would you like further details or have any questions?

Here are five related questions to extend your understanding:

1. How do you derive the general formula for completing the square?
2. What are the steps to convert the general equation of a circle into standard form?
3. How do you determine the distance between two points on a coordinate plane?
4. How does the radius affect the size of the circle on the graph?
5. How would you graph a circle that is not centered at the origin?

Tip: Completing the square is a useful technique not only for circles but also for solving quadratic equations!