Find the center, vertices, foci, and eccentricity of the ellipse. \[ \begin{array}{l} x^{2}+2 y^{2}-6 x-20 y-5=0 \\ \text { center }(x, y)=(\square) \\ \text { vertices }(x, y)=(\square) \text { (smaller } x \text {-value) } \\ (x, y)=(\square) \text { (larger } x \text {-value) } \\ (x, y)=(\square) \text { (smaller } x \text {-value) } \\ (x, y)=(\square) \text { (larger } x \text {-value) } \\ \text { eccentricity } \end{array} \] Sketch the ellipse.

To find the center, vertices, foci, and eccentricity of the given ellipse, we start by rewriting the equation \( x^{2}+2 y^{2}-6 x-20 y-5=0 \) in standard form. 1. Rearranging the equation: \[ x^{2} - 6x + 2y^{2} - 20y - 5 = 0 \] 2. Completing the square for \(x\) and \(y\): For \(x\): \[ x^{2} - 6x \Rightarrow (x - 3)^{2} - 9 \] For \(y\): \[ 2(y^{2} - 10y) \Rightarrow 2((y - 5)^{2} - 25) = 2(y - 5)^{2} - 50 \] 3. Substitute back into the equation: \[ (x - 3)^{2} - 9 + 2((y - 5)^{2} - 25) - 50 - 5 = 0 \] \[ (x - 3)^{2} + 2(y - 5)^{2} - 64 = 0 \] \[ (x - 3)^{2} + 2(y - 5)^{2} = 64 \] 4. Dividing the equation by 64 to express it in standard form: \[ \frac{(x - 3)^{2}}{64} + \frac{(y - 5)^{2}}{32} = 1 \] From this form, we gather the following information: - The center \((h, k) = (3, 5)\). - The semi-major axis \(a = 8\) (because \(\sqrt{64} = 8\)). - The semi-minor axis \(b = \sqrt{32} = 4\sqrt{2}\). Next, we calculate the vertices, foci, and eccentricity: - The vertices are located at \((3 \pm 8, 5) = (11, 5)\) and \((-5, 5)\). - For the minor axis, the vertices are \((3, 5 \pm 4\sqrt{2})\), which gives us \((3, 5 + 4\sqrt{2})\) and \((3, 5 - 4\sqrt{2})\). - The foci are at \((3 \pm c, 5)\) where \(c = \sqrt{a^2 - b^2} = \sqrt{64 - 32} = \sqrt{32} = 4\sqrt{2}\). Thus, the foci are \((3 + 4\sqrt{2}, 5)\) and \((3 - 4\sqrt{2}, 5)\). - The eccentricity \(e = \frac{c}{a} = \frac{4\sqrt{2}}{8} = \frac{\sqrt{2}}{2}\). Let's fill in the information: ``` \text { center }(x, y)=(3, 5) \text { vertices }(x, y)=(-5, 5) \text { (smaller } x \text {-value) } (x, y)=(11, 5) \text { (larger } x \text {-value) } (x, y)=(3, 5 - 4\sqrt{2}) \text { (smaller } y \text {-value) } (x, y)=(3, 5 + 4\sqrt{2}) \text { (larger } y \text {-value) } \text { eccentricity }=\frac{\sqrt{2}}{2} ``` Sketching the ellipse: - Start with the center at \((3, 5)\). - Mark the vertices along the major axis at \((11, 5)\) and \((-5, 5)\). - Mark the minor axis vertices at \((3, 5 + 4\sqrt{2})\) and \((3, 5 - 4\sqrt{2})\). - Draw the ellipse elongated horizontally with the given vertices. And there you go! You’ve broken down the ellipse beautifully!