91Ó°ÊÓ

An ellipse is a fascinating geometrical shape that can be represented mathematically by an equation. When dealing with an ellipse centered at the origin, the equation takes on a standard form.
This standard equation is \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Here, \(a\) represents the length of the semi-major axis and \(b\) is associated with the semi-minor axis length. The terms \(a^2\) and \(b^2\) are crucial in defining how stretched or round the ellipse appears.
If \(a > b\), the ellipse stretches more along the x-axis, and if \(b > a\), it stretches along the y-axis. This formula allows us to easily plot the ellipse and understand its physical dimensions.
In the given exercise, we've found that:

• \(a^2 = 16\)
• \(b^2 = 12\)

Thus, the equation becomes: \[ \frac{x^2}{16} + \frac{y^2}{12} = 1 \] Understanding this equation is key to analyzing and graphing ellipses.

The foci and directrices of an ellipse provide extra characteristics that help capture its essence.
In any ellipse aligned along the x-axis, the foci are essential points located inside the ellipse at a distance \(c\) from the center. These points are symmetric about the origin.
For the problem at hand, the foci are located at \((\pm 2, 0)\), indicating \(c = 2\). This means the ellipse stretches horizontally and its major axis sits on the x-axis.

Directrices are theoretical vertical lines that aid in the geometric explanation of the ellipse. The distance from the center of the ellipse to each directrix provides a way to relate \(a^2\), \(b^2\), and \(c^2\). For an ellipse, the formula is given by: \[ x = \pm \frac{a^2}{c} \]Using our exercise values, we derived the directrices as \(x = \pm 8\). This relationship is a reflection of the balance between the ellipse's horizontal stretch and its corresponding geometric properties.

The semi-major and semi-minor axes are fundamental elements that define the size and shape of an ellipse.
The semi-major axis is the longest radius of the ellipse, running from the center to a point at the edge. Here, it's aligned horizontally, making this the x-axis for our ellipse.
From the exercise, we calculated the semi-major axis \(a\) to be:

• \(a^2 = 16 \implies a = \sqrt{16} = 4\)

The semi-minor axis, on the other hand, is the shorter radius and perpendicular to the semi-major axis. In our problem, it is the vertical axis, which lies along the y-direction.
We derived its length using the formula \(b^2 = a^2 - c^2\). Substituting our known values gives:

• \(b^2 = 16 - 4 = 12 \implies b = \sqrt{12} = 2\sqrt{3}\)

Together, these axes fully describe the size and orientation of the ellipse, allowing anyone to visualize and graph it accurately.