Determine the general equation of : center(3,-4) and area is 16pi

To determine the general equation of a circle given its center and area, we can follow these steps:

### Step 1: Understand the general equation of a circle.
The general equation of a circle is given by:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle, and \(r\) is the radius.

### Step 2: Identify the center of the circle.
From the problem, the center of the circle is \((3, -4)\), so \(h = 3\) and \(k = -4\).

### Step 3: Calculate the radius from the area.
The area \(A\) of a circle is given by:
\[
A = \pi r^2
\]
We know the area is \(16\pi\), so:
\[
16\pi = \pi r^2
\]
Dividing both sides by \(\pi\):
\[
r^2 = 16
\]
Taking the square root of both sides:
\[
r = 4
\]

### Step 4: Plug the values into the general equation.
Now that we have \(r = 4\), \(h = 3\), and \(k = -4\), we can substitute these into the general equation of the circle:
\[
(x - 3)^2 + (y + 4)^2 = 4^2
\]
Simplifying:
\[
(x - 3)^2 + (y + 4)^2 = 16
\]

### Final Answer:
The general equation of the circle is:
\[
(x - 3)^2 + (y + 4)^2 = 16
\]

Would you like more details on this process or have any questions?

Here are some related questions:
1. How would the equation change if the area were \(25\pi\)?
2. What would the equation be if the circle were centered at the origin?
3. How do you determine the radius from the circumference instead of the area?
4. Can you derive the equation if the diameter is given instead of the area?
5. How does the equation change if the circle is translated horizontally or vertically?

**Tip:** Always verify the radius by calculating it from the area before plugging it into the general equation of the circle.

Determine the general equation of a circle with center (3, -4) and area is 16pi.

Learn how to find the general equation of a circle with center (3, -4) and area 16π. Step-by-step solution included. Suitable for high school students studying geometry and circle equations.

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