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When you graph an ellipse, it is helpful to start by understanding its standard equation form. An ellipse can be represented by the equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) if the center is at the origin \((0,0)\). Here, \(a\) and \(b\) are the distances from the center to the vertices along the x-axis and y-axis respectively, which determine the ellipse's shape and orientation.
In the given example, \(\frac{x^{2}}{25} + \frac{y^{2}}{64} = 1\), we find that \(b^{2} = 64\) is larger than \(a^{2} = 25\). This tells us that the ellipse is a vertical one, as the larger denominator is associated with \(y\), indicating the major axis runs parallel to the y-axis.
To graph this ellipse, we:

• Identify the center, which is at \((0,0)\).
• Move \(a\) units to the left and right of the y-axis to plot the vertices of the minor axis.
• Move \(b\) units up and down, parallel to the y-axis, to plot the vertices of the major axis.

Once these key points are plotted, draw a smooth curve connecting them in an oval shape to complete the graph. Don't forget to label the foci, which we will explore further.

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. An ellipse, like the one we're working with, forms when the plane intersects the cone at an angle that is less steep than the cone's side but does not pass through the base.
Ellipses have unique properties that set them apart:

• The sum of the distances from any point on the ellipse to two fixed points, called foci, is constant.
• Ellipses have two axes: the major axis, which is the longest diameter of the ellipse, and the minor axis, the shortest.

In practice, understanding the positioning and geometry of conic sections helps in graphing and analyzing various curves in mathematics and physics. For ellipses, the foci play a critical role in their definition and properties, such as reflecting properties of light and sound.

The foci of an ellipse are two fixed points located on the major axis equidistant from the center. They are crucial in defining the shape and properties of an ellipse. Unlike the center of a circle, which is a single point that doesn't change, ellipses have these two points that dictate their distinct shape.
To find the foci in our example, use the formula \(c = \sqrt{b^{2} - a^{2}}\). Here \(a = 5\) and \(b = 8\), so \(c = \sqrt{64 - 25} = \sqrt{39}\). This gives us the foci at \((0, \pm \sqrt{39})\), as our ellipse has its major axis aligned vertically.
This relationship between the semi-major axis \((b)\), semi-minor axis \((a)\), and the foci \((c)\) is a fundamental characteristic of ellipses and is essential for both understanding and graphing them. By knowing how to calculate and locate the foci, you can better appreciate the mathematical elegance and applications of ellipses.