1. Problem: Analyze the ellipses in Figures 1 and 2 and find the center, vertices, co-vertices, foci, lengths of axes, and equations for each. **Figure 1:** 1. Center: The ellipse is centered at the origin, so the center is $\boxed{(0,0)}$. 2. Vertices: The ellipse crosses the x-axis at $\pm7$, so the vertices are $\boxed{(7,0)}$ and $\boxed{(-7,0)}$. 3. Co-vertices: The ellipse crosses the y-axis at $\pm5$, so the co-vertices are $\boxed{(0,5)}$ and $\boxed{(0,-5)}$. 4. Foci: For an ellipse centered at origin with major axis along x-axis, the foci are at $\pm c$ where $$c=\sqrt{a^2 - b^2}=\sqrt{7^2 - 5^2}=\sqrt{49-25}=\sqrt{24}=2\sqrt{6}$$ so foci are at $\boxed{(2\sqrt{6},0)}$ and $\boxed{(-2\sqrt{6},0)}$. 5. Lengths: - Major axis length = $2a=2\times7=14$ - Minor axis length = $2b=2\times5=10$ 6. Standard equation: Since major axis is along x-axis, $$\frac{x^2}{7^2}+\frac{y^2}{5^2}=1 \implies \frac{x^2}{49}+\frac{y^2}{25}=1$$ General equation by multiplying both sides by 1225: $$25x^2 + 49 y^2 = 1225$$ **Figure 2:** 1. Center: Again, centered at origin $\boxed{(0,0)}$. 2. Vertices: Vertices are along the y-axis at approximately $\pm5$, so vertices are $\boxed{(0,5)}$ and $\boxed{(0,-5)}$. 3. Co-vertices: Co-vertices along x-axis approximately at $\pm3$, so co-vertices are $\boxed{(3,0)}$ and $\boxed{(-3,0)}$. 4. Foci: For ellipse with major axis vertical, $$c=\sqrt{a^2-b^2}=\sqrt{5^2 - 3^2}=\sqrt{25 - 9} = \sqrt{16} = 4$$ Foci are located at $\boxed{(0,4)}$ and $\boxed{(0,-4)}$. 5. Lengths: - Major axis length = $2a=2\times5=10$ - Minor axis length = $2b=2\times3=6$ 6. Standard equation: Major axis along y-axis, $$\frac{x^2}{3^2} + \frac{y^2}{5^2} =1 \implies \frac{x^2}{9} + \frac{y^2}{25} =1$$ General equation by multiplying by 225: $$25x^2 + 9y^2 = 225$$ Final answers summarized: - Figure 1: - Center: $(0,0)$ - Vertices: $(\pm7,0)$ - Co-vertices: $(0,\pm5)$ - Foci: $(\pm 2\sqrt{6},0)$ - Major axis length: 14 - Minor axis length: 10 - Standard Eq: $\frac{x^2}{49}+\frac{y^2}{25}=1$ - General Eq: $25x^2 + 49 y^2 = 1225$ - Figure 2: - Center: $(0,0)$ - Vertices: $(0,\pm5)$ - Co-vertices: $(\pm3,0)$ - Foci: $(0,\pm4)$ - Major axis length: 10 - Minor axis length: 6 - Standard Eq: $\frac{x^2}{9} + \frac{y^2}{25} =1$ - General Eq: $25x^2 + 9 y^2 = 225$