91Ó°ÊÓ

The semi-major and semi-minor axes are crucial to the shape of an ellipse. These are the 'radii' of the ellipse, measuring half the length of the ellipse's longest and shortest diameters respectively. The semi-major axis runs from the center to the farthest edge and is denoted as 'a'. Conversely, the semi-minor axis, denoted as 'b', stretches from the center to the closest edge.

Using the given equation, \(x^2 / 16 + y^2 / 25 = 1\), we can deduce that \(a^2 = 16\) and \(b^2 = 25\). Hence, \(a = 4\) and \(b = 5\), signifying that the semi-minor axis is 4 units in length while the semi-major axis is 5 units long. This difference in the axes' lengths gives the ellipse its characteristic oval shape.

The equation of an ellipse in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) provides us with a blueprint of this geometric figure on a coordinate plane. Here, \(a\) and \(b\) symbolize the lengths of the semi-major and semi-minor axes respectively. If \(b > a\), the ellipse stretches more along the y-axis, making it taller. Conversely, if \(a > b\), the ellipse is wider, stretching more along the x-axis.

The given equation \(\frac{x^2}{16} + \frac{y^2}{25} = 1\) tells us the ellipse’s exact shape and orientation on the graph. By rewriting it as \(x^2 + y^2 \frac{16}{25} = 16\), we can see more clearly that \(b\) is associated with the y-coordinate and is greater than \(a\), indicating a taller ellipse.

Eccentricity, denoted as \(e\), measures an ellipse's deviation from being a perfect circle. It is a dimensionless number that ranges from 0 (a circle) to just below 1 (a highly stretched ellipse). To calculate it, use the formula \(e = \sqrt{1 - \frac{a^2}{b^2}}\), where \(a\) is the length of the semi-minor axis and \(b\) is the length of the semi-major axis.

For the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\), we determined that \(a = 4\) and \(b = 5\). Plugging these values into the formula, we get \(e = \sqrt{1 - \frac{16}{25}}\). Simplifying the square root, the eccentricity is thus \(\frac{3}{5}\). This indicates that the ellipse is moderately elongated, being neither a circle (\(e = 0\)) nor too elongated (with a high \(e\) value approaching 1).