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The standard form of an ellipse equation is a blueprint that dictates how the ellipse is shaped and positioned in the coordinate plane. An ellipse can be thought of as a stretched circle with two axes of symmetry - the major and minor axes.

The standard form is expressed as \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\) if the major axis is horizontal, or \(\frac{(y-k)^{2}}{a^{2}} + \frac{(x-h)^{2}}{b^{2}} = 1\) if the major axis is vertical. Here, \(h, k\) represents the center of the ellipse, while \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

• Center (h, k): The point around which the ellipse is symmetrically drawn.
• Semi-major axis (a): Half of the longest diameter of the ellipse.
• Semi-minor axis (b): Half of the shortest diameter of the ellipse.

For the given exercise, rearranging the given equation to fit the standard form made it clear that the center is at (-2, 3), with semi-major axis \(a = 4\) and semi-minor axis \(b = 1\). This understanding is crucial to correctly plotting an ellipse and determines the shape and location of the graph.

The foci (plural of focus) of an ellipse are two fixed points located along the major axis, equidistant from the center, which have a sum of distances to any point on the ellipse that is constant. This unique pair of points plays a vital role in the ellipse's definition and properties.

The calculation of the ellipse foci involves finding the distance \(c\) from the center to each focus. To find \(c\), you can use the equation \(c = \sqrt{\lvert a^{2} - b^{2} \rvert}\). In this equation, \(a\) is the length of the semi-major axis and \(b\) is the length of the semi-minor axis. Notice that in the given exercise, since \(a > b\), we have \(c = \sqrt{16 - 1} = \sqrt{15}\), meaning the foci are \(\sqrt{15}\) units away from the center along the major axis. The precise location of the foci in the coordinate plane will be (-2 ± \sqrt{15}, 3), reflecting their symmetry about the center of the ellipse.

Understanding the major and minor axes of an ellipse is key to graphing it correctly. The major axis is the longest diameter of the ellipse, stretching from one end to the other through the center, while the minor axis is the shortest diameter, also passing through the center, perpendicular to the major axis.

The lengths of the major and minor axes are respectively given by \(2a\) and \(2b\), where \(a\) and \(b\) are the semi-major and semi-minor axes as defined in the standard form of the ellipse equation. When \(a > b\), the ellipse is wider than it is tall, indicating a horizontal major axis. Conversely, if \(b > a\), the major axis would be vertical.

In our exercise, we determined that \(a = 4\) and \(b = 1\), which means the length of the major axis is 8 units and the length of the minor axis is 2 units. Knowing these dimensions is fundamental to plotting the ellipse on the graph as they dictate how far to draw the ellipse from the center along each axis.