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When talking about ellipses, the term "standard form" refers to the equation format that describes an ellipse on a Cartesian coordinate plane. An ellipse's standard form equation looks like this:

• Horizontal Ellipse: \( \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \)
• Vertical Ellipse: \( \frac{(x-h)^{2}}{b^{2}} + \frac{(y-k)^{2}}{a^{2}} = 1 \)

Here, \((h, k)\) represents the center of the ellipse, and \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. Standard form is incredibly useful because it allows us to quickly identify these axis lengths. In our exercise, the ellipse is centered at the origin, \((0,0)\), so the equation simplifies to \( \frac{x^{2}}{64} + \frac{y^{2}}{28} = 1 \), with \(a^{2} = 64\) and \(b^{2} = 28\). Both coefficients provide crucial information about the ellipse's size and orientation on the graph.

The semi-major axis of an ellipse is the longest radius that extends from the center to the outer edge along the ellipse's major axis. The length of the semi-major axis is given by \(a\) in the standard form, where \(a^{2}\) is always the larger of the two denominators. In our example, since \(64\) is greater than \(28\), we find \(a\) by calculating \(\sqrt{64} = 8\). This means that the length of the semi-major axis is 8.The orientation of the semi-major axis also tells us about the ellipse's orientation. If \(a\) is associated with \(x\), the ellipse stretches more in the horizontal direction; if associated with \(y\), it stretches vertically. In our case, the larger denominator is under \(x^{2}\), making the ellipse horizontally elongated.

The semi-minor axis of an ellipse is the shortest radius, extending from the center to the edge along the minor axis. In the equation's standard form, the semi-minor axis is denoted by \(b\), where \(b^{2}\) is the smaller denominator. In our given equation, \(28\) is the smaller value, which leads us to compute \(b\) as \(\sqrt{28}\).Breaking further into the semi-minor axis calculation, \(b = \sqrt{28} = 2\sqrt{7}\) provides us with a depiction of how broad or narrow the ellipse is along the minor axis. As \(b\) works in tandem with \(a\) to outline the ellipse's shape, it's important to understand each axis's role in graphing the ellipse accurately.