An equation of an ellipse is given. \( \begin{array}{ll}\text { (a) Identify the center of the ellipse. } & \text { (b) Determine the value of } a, \\ \text { (c) Determine the value of } b . & \text { (d) Identify the vertices. } \\ \begin{array}{ll}\text { (e) Identify the endpoints of the minor axis. } & \text { (f) Identify the foci. } \\ \text { (g) Determine the length of the major axis. } & \text { (h) Determine the length of the minor axis. } \\ \text { (i) Graph the ellipse. } \\ \text { Express numbers in exact, simplest form. } \\ \frac{x^{2}}{49}+\frac{y^{2}}{16}=1\end{array}\end{array}>. l \)

The center of the ellipse described by the equation \(\frac{x^{2}}{49}+\frac{y^{2}}{16}=1\) is located at the coordinate point (0, 0), which is considered the origin. The values of \(a\) and \(b\) correspond to the square roots of the denominators of the fractions in the equation. Specifically, \(a = 7\) (since \(a^2 = 49\)) and \(b = 4\) (since \(b^2 = 16\)). When determining the vertex points of the ellipse, they lie at \( ( \pm a, 0 )\) which is \( ( \pm 7, 0 )\), while the endpoints of the minor axis are found at \( (0, \pm b)\), giving us the points \( (0, \pm 4) \). To find the foci, we can use the formula \(c = \sqrt{a^2 - b^2}\), so \(c = \sqrt{49 - 16} = \sqrt{33}\), making the foci at \( (\pm \sqrt{33}, 0) \). The length of the major axis is \(2a = 14\) and the minor axis length is \(2b = 8\). Plotting the ellipse on a coordinate system would show it stretching wider along the x-axis (major axis) and shorter along the y-axis (minor axis). Fun fact: Ellipses can be seen in nature, such as in the orbits of planets or the shape of some fruits, adding a bit of beauty to mathematical study!