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.

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.