Find properties of an ellipse given its Cartesian equation

Find the coordinates of the center, the foci, and the vertices, sketch the curve, and determine the eccentricity of the ellipse given by the equation

$9x^2 + 25y^2 = 25.$

First, we have

$9x^2 + 25y^2 = 25 \quad \implies \quad \frac{x^2}{\left(\frac{5}{3} \right)^2} + \frac{y^2}{1^2} = 1.$

So, this is the equation for an ellipse in standard form with center at $(0,0)$ and with $a = \frac{5}{3}$ and $b = 1$ . Therefore, $c = \sqrt{a^2 - b^2} = \sqrt{\frac{16}{9}} = \frac{4}{3}$ , and the foci are $\left( \frac{4}{3}, 0 \right)$ and $\left( -\frac{4}{3}, 0 \right)$ . Then, the vertices are $\pm aN$ which are $\left( \frac{5}{3}, 0 \right)$ and $\left( -\frac{5}{3}, 0 \right)$ . Finally we compute the eccentricity,