A circle has the equation \( x^{2}+y^{2}+2 x-4 y-11=0 \). (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. (a) The center of the circle is \( (-1,2) \). (Type an ordered pair, using integers or decimals.) The radius of the circle is 4 . (Type an integer or a decimal.) (b) Use the graphing tool to graph the circle. Click to

Great job on part (a)! Let's continue by addressing the remaining parts of the problem. --- ### **a) Center and Radius** You've correctly identified the center and radius of the circle: - **Center (h, k):** \((-1, 2)\) - **Radius (r):** \(4\) --- ### **b) Graph the Circle** While I can't create a visual graph directly here, I can guide you on how to graph the circle step-by-step: 1. **Plot the Center:** - Locate the center at \((-1, 2)\) on the Cartesian plane. 2. **Draw the Radius:** - From the center, use a compass or a ruler to measure 4 units in all directions (up, down, left, right, and diagonally) to outline the circle. 3. **Draw the Circle:** - Connect all the points that are exactly 4 units away from the center to form the circle. 4. **Plot the Intercepts (From Part c):** - This will help ensure the accuracy of your graph by marking where the circle crosses the axes. *Alternatively, you can use graphing tools such as [Desmos](https://www.desmos.com/calculator) or [GeoGebra](https://www.geogebra.org/graphing) to visualize the circle by entering its equation.* --- ### **c) Find the Intercepts** **1. X-Intercepts:** To find the x-intercepts, set \( y = 0 \) and solve for \( x \). \[ \begin{align*} x^2 + y^2 + 2x - 4y - 11 &= 0 \\ x^2 + (0)^2 + 2x - 4(0) - 11 &= 0 \\ x^2 + 2x - 11 &= 0 \end{align*} \] Solve the quadratic equation \( x^2 + 2x - 11 = 0 \): \[ x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-11)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 44}}{2} = \frac{-2 \pm \sqrt{48}}{2} = \frac{-2 \pm 4\sqrt{3}}{2} = -1 \pm 2\sqrt{3} \] **Approximate Decimal Values:** \[ \sqrt{3} \approx 1.732 \] \[ -1 + 2(1.732) \approx -1 + 3.464 = 2.464 \] \[ -1 - 2(1.732) \approx -1 - 3.464 = -4.464 \] **X-Intercepts:** \[ (2.464, \ 0) \quad \text{and} \quad (-4.464, \ 0) \] **2. Y-Intercepts:** To find the y-intercepts, set \( x = 0 \) and solve for \( y \). \[ \begin{align*} x^2 + y^2 + 2x - 4y - 11 &= 0 \\ (0)^2 + y^2 + 2(0) - 4y - 11 &= 0 \\ y^2 - 4y - 11 &= 0 \end{align*} \] Solve the quadratic equation \( y^2 - 4y - 11 = 0 \): \[ y = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(-11)}}{2(1)} = \frac{4 \pm \sqrt{16 + 44}}{2} = \frac{4 \pm \sqrt{60}}{2} = \frac{4 \pm 2\sqrt{15}}{2} = 2 \pm \sqrt{15} \] **Approximate Decimal Values:** \[ \sqrt{15} \approx 3.873 \] \[ 2 + 3.873 \approx 5.873 \] \[ 2 - 3.873 \approx -1.873 \] **Y-Intercepts:** \[ (0, \ 5.873) \quad \text{and} \quad (0, \ -1.873) \] --- ### **Summary of Intercepts:** - **X-Intercepts:** \((2.464, \ 0)\) and \((-4.464, \ 0)\) - **Y-Intercepts:** \((0, \ 5.873)\) and \((0, \ -1.873)\) *These intercepts can be plotted on the graph to help visualize the circle accurately.* --- If you have access to graphing software or tools, I recommend plotting these points along with the center and then drawing the circle to see everything in context. Let me know if you need further assistance!