The equation of a circle is given below. Identify the center and radius. Then graph the circle. $$ (x-2)^{2}+(y-5)^{2}=16 $$

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UpStudy AI · Accepted Answer

To identify the center and radius of the circle given by the equation

$$
(x-2)^{2}+(y-5)^{2}=16,
$$

we can compare it to the standard form of the equation of a circle, which is

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

where $$(h, k)$$ is the center of the circle and $$r$$ is the radius.

### Step 1: Identify the center and radius

From the given equation:

- The term $$(x-2)^{2}$$ indicates that $$h = 2$$.
- The term $$(y-5)^{2}$$ indicates that $$k = 5$$.
- The right side of the equation is $$16$$, which means $$r^{2} = 16$$. Therefore, the radius $$r$$ is

$$
r = \sqrt{16} = 4.
$$

### Summary of the center and radius
- Center: $$(2, 5)$$
- Radius: $$4$$

### Step 2: Graph the circle

To graph the circle, we will plot the center at $$(2, 5)$$ and then use the radius to mark points that are $$4$$ units away from the center in all directions (up, down, left, right).

1. **Center**: Plot the point $$(2, 5)$$.
2. **Radius**: From the center:
   - Move $$4$$ units up to $$(2, 9)$$.
   - Move $$4$$ units down to $$(2, 1)$$.
   - Move $$4$$ units left to $$(-2, 5)$$.
   - Move $$4$$ units right to $$(6, 5)$$.

Now, I will graph the circle based on this information.

### Graphing the Circle

Let's proceed to graph the circle.

To visualize the circle, I will plot the center and the points determined by the radius.

Here is the graph of the circle:

- Center at $$(2, 5)$$
- Points at $$(2, 9)$$, $$(2, 1)$$, $$(-2, 5)$$, and $$(6, 5)$$

Now, I will create the graph.

(Note: As a text-based AI, I cannot create visual graphs directly, but you can use graphing software or graph paper to plot the points and draw the circle.)

The circle will be centered at $$(2, 5)$$ with a radius of $$4$$.
The center of the circle is at $$(2, 5)$$ and the radius is $$4$$.

The equation of a circle is given below. Identify the center and radius. Then graph the circle. \( (x-2)^{2}+(y-5)^{2}=16 \)

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