A circle has the equation $$ x^{2}+y^{2}+2 x-4 y-11= $$ (a) Find the center (h,k) and radius $$ r $$ of the circle (b) Graph the circle. (c) Find the intercepts, if any, of the graph.

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To solve the problem, we will follow these steps:

### Part (a): Find the center $$(h, k)$$ and radius $$r$$ of the circle

The equation of the circle is given as:

$$
x^{2} + y^{2} + 2x - 4y - 11 = 0
$$

We will rearrange this equation into the standard form of a circle, which is:

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

1. **Rearranging the equation:**
   $$
   x^{2} + 2x + y^{2} - 4y = 11
   $$

2. **Completing the square for $$x$$:**
   - Take the coefficient of $$x$$ (which is 2), halve it (1), and square it (1).
   - Add and subtract 1:
   $$
   (x^{2} + 2x + 1) - 1
   $$

3. **Completing the square for $$y$$:**
   - Take the coefficient of $$y$$ (which is -4), halve it (-2), and square it (4).
   - Add and subtract 4:
   $$
   (y^{2} - 4y + 4) - 4
   $$

4. **Substituting back into the equation:**
   $$
   (x + 1)^{2} - 1 + (y - 2)^{2} - 4 = 11
   $$
   $$
   (x + 1)^{2} + (y - 2)^{2} - 5 = 11
   $$
   $$
   (x + 1)^{2} + (y - 2)^{2} = 16
   $$

Now we can identify the center and radius:
- Center $$(h, k) = (-1, 2)$$
- Radius $$r = \sqrt{16} = 4$$

### Part (b): Graph the circle

To graph the circle, we will plot the center at $$(-1, 2)$$ and draw a circle with a radius of 4.

### Part (c): Find the intercepts, if any, of the graph

1. **Finding the x-intercepts:**
   Set $$y = 0$$ in the original equation:
   $$
   x^{2} + 0^{2} + 2x - 4(0) - 11 = 0
   $$
   $$
   x^{2} + 2x - 11 = 0
   $$

2. **Finding the y-intercepts:**
   Set $$x = 0$$ in the original equation:
   $$
   0^{2} + y^{2} + 2(0) - 4y - 11 = 0
   $$
   $$
   y^{2} - 4y - 11 = 0
   $$

Now, we will solve these equations to find the intercepts.

Let's perform the calculations for the x-intercepts and y-intercepts.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$y^{2}-4y-11=0$$
- step1: Solve using the quadratic formula:
  $$y=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\left(-11\right)}}{2}$$
- step2: Simplify the expression:
  $$y=\frac{4\pm \sqrt{60}}{2}$$
- step3: Simplify the expression:
  $$y=\frac{4\pm 2\sqrt{15}}{2}$$
- step4: Separate into possible cases:
  $$\begin{align}&y=\frac{4+2\sqrt{15}}{2}\&y=\frac{4-2\sqrt{15}}{2}\end{align}$$
- step5: Simplify the expression:
  $$\begin{align}&y=2+\sqrt{15}\&y=\frac{4-2\sqrt{15}}{2}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&y=2+\sqrt{15}\&y=2-\sqrt{15}\end{align}$$
- step7: Rewrite:
  $$y_{1}=2-\sqrt{15},y_{2}=2+\sqrt{15}$$
Solve the equation $$ x^{2}+2x-11=0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$x^{2}+2x-11=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{-2\pm \sqrt{2^{2}-4\left(-11\right)}}{2}$$
- step2: Simplify the expression:
  $$x=\frac{-2\pm \sqrt{48}}{2}$$
- step3: Simplify the expression:
  $$x=\frac{-2\pm 4\sqrt{3}}{2}$$
- step4: Separate into possible cases:
  $$\begin{align}&x=\frac{-2+4\sqrt{3}}{2}\&x=\frac{-2-4\sqrt{3}}{2}\end{align}$$
- step5: Simplify the expression:
  $$\begin{align}&x=-1+2\sqrt{3}\&x=\frac{-2-4\sqrt{3}}{2}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=-1+2\sqrt{3}\&x=-1-2\sqrt{3}\end{align}$$
- step7: Rewrite:
  $$x_{1}=-1-2\sqrt{3},x_{2}=-1+2\sqrt{3}$$
### Part (a): Center and Radius
- Center $$(h, k) = (-1, 2)$$
- Radius $$r = 4$$

### Part (b): Graph the Circle
To graph the circle, plot the center at $$(-1, 2)$$ and draw a circle with a radius of 4 units. The circle will extend from $$(-5, 2)$$ to $$(3, 2)$$ horizontally and from $$(-1, -2)$$ to $$(-1, 6)$$ vertically.

### Part (c): Finding the Intercepts

1. **X-Intercepts:**
   The x-intercepts are found by solving the equation:
   $$
   x^{2} + 2x - 11 = 0
   $$
   The solutions are:
   $$
   x_{1} = -1 - 2\sqrt{3}, \quad x_{2} = -1 + 2\sqrt{3}
   $$

2. **Y-Intercepts:**
   The y-intercepts are found by solving the equation:
   $$
   y^{2} - 4y - 11 = 0
   $$
   The solutions are:
   $$
   y_{1} = 2 - \sqrt{15}, \quad y_{2} = 2 + \sqrt{15}
   $$

### Summary of Intercepts
- **X-Intercepts:** 
  - $$x_{1} = -1 - 2\sqrt{3} \approx -4.464$$
  - $$x_{2} = -1 + 2\sqrt{3} \approx 2.464$$

- **Y-Intercepts:**
  - $$y_{1} = 2 - \sqrt{15} \approx -0.872$$
  - $$y_{2} = 2 + \sqrt{15} \approx 4.872$$

This completes the solution for the circle's center, radius, graph, and intercepts.
The circle has its center at $$(-1, 2)$$ and a radius of 4. It intercepts the x-axis at approximately $$-4.464$$ and $$2.464$$, and the y-axis at approximately $$-0.872$$ and $$4.872$$.

A circle has the equation \( x^{2}+y^{2}+2 x-4 y-11= \) (a) Find the center (h,k) and radius \( r \) of the circle (b) Graph the circle. (c) Find the intercepts, if any, of the graph.

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