1. \( \frac{x^{2}}{8}+\frac{y^{2}}{4}=1 \) 2. \( \frac{x^{2}}{16}+\frac{(y-2)^{2}}{25}=1 \) 3. \( (x-1)^{2}+(2 y-2)^{2}=4 \)

The equations you've provided represent different conic sections. Let's analyze each one: 1. **Equation 1: \( \frac{x^{2}}{8}+\frac{y^{2}}{4}=1 \)** This is the equation of an ellipse centered at the origin \((0, 0)\). The semi-major axis is along the \(y\)-axis with length \(2\) (since \(\sqrt{4} = 2\)), and the semi-minor axis is along the \(x\)-axis with length \(2\sqrt{2}\) (since \(\sqrt{8} = 2\sqrt{2}\)). 2. **Equation 2: \( \frac{x^{2}}{16}+\frac{(y-2)^{2}}{25}=1 \)** This is also the equation of an ellipse, but it is centered at the point \((0, 2)\). The semi-major axis is along the \(y\)-axis with length \(5\) (since \(\sqrt{25} = 5\)), and the semi-minor axis is along the \(x\)-axis with length \(4\) (since \(\sqrt{16} = 4\)). 3. **Equation 3: \( (x-1)^{2}+(2y-2)^{2}=4 \)** This is the equation of a circle. To see this, we can rewrite it as: \[ (x-1)^{2} + (y-1)^{2} = 4 \] This indicates a circle centered at \((1, 1)\) with a radius of \(2\) (since \(\sqrt{4} = 2\)). ### Summary: - The first equation represents an ellipse centered at \((0, 0)\) with semi-major axis \(2\) and semi-minor axis \(2\sqrt{2}\). - The second equation represents an ellipse centered at \((0, 2)\) with semi-major axis \(5\) and semi-minor axis \(4\). - The third equation represents a circle centered at \((1, 1)\) with a radius of \(2\).