91Ó°ÊÓ

The foci of an ellipse are two special points located along the major axis. These points help define the ellipse's shape and are critical in its geometric properties.
The sum of the distances from any point on the ellipse to the foci is constant.
In simpler terms, this means if you pick any spot on the ellipse, the two lines drawn to each focus always add up to the same total length. This constant distance proves essential in the construction and definition of an ellipse.To locate the foci, you can use the formula \(c = \sqrt{b^2 - a^2}\), where \(c\) is the distance from the center to each focus. Here, \(a\) and \(b\) are the semi-axis lengths. In our solved problem, we found that \(c = \sqrt{5}\), meaning the foci are \((0, \sqrt{5})\) and \((0, -\sqrt{5})\).
This positions them along the vertical axis of the ellipse.

The standard form of the ellipse equation helps us understand its position, shape, and orientation. It looks like this:\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]Here:

• \((h, k)\) is the center of the ellipse.
• \(a\) and \(b\) are the semi-axes' lengths.

In the problem uses, the equation \(9x^2 + 4y^2 = 36\) initially wasn't in standard form.
By manipulating it, we converted it to \(\frac{x^2}{4} + \frac{y^2}{9} = 1\). This places its center at the origin \((0, 0)\) with \(a = 2\) and \(b = 3\).
Converting equations to this format simplifies identifying essential features of the ellipse, like its axes and foci.

An ellipse stretches along two main directions: the major and minor axes.
The major axis is the longest diameter of the ellipse, while the minor axis is the shortest.For the standard form, if \(b > a\), the major axis is vertical, as seen in this ellipse centered at the origin:

• Major Axis: Length \(2b\) units, which is 6 units vertically.
• Minor Axis: Length \(2a\) units, which is 4 units horizontally.

Vertices and co-vertices lie along these axes:
- Vertices are points along the major axis at \((0, 3)\) and \((0, -3)\).
- Co-vertices sit on the minor axis at \((2, 0)\) and \((-2, 0)\).
Understanding these axes gives insight into the ellipse's overall shape and positioning in graphing.