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In the geometry of ellipses, **vertices** are the furthest points along the main axis of the ellipse. For the given equation of an ellipse \[ \frac{x^2}{2} + \frac{y^2}{5} = 1 \],we see that the ellipse is vertical based on the greater denominator being under \(y^2\). This vertical orientation tells us that the **vertices** are aligned along the y-axis.

• The vertices of such a vertical ellipse can be located at \((0, \pm b)\), where \(b\) denotes the length of the semi-major axis.
• Given \(b = \sqrt{5} \approx 2.24\), the vertices are \((0, -2.24)\) and \((0, 2.24)\).

These points are critical because they help determine the "height" of the ellipse visually and mathematically.

The **foci** (plural of focus) of an ellipse are two special points on the major axis. The sum of the distances from these points to any point on the ellipse is a constant. For ellipse calculations, the distance from the center to each focus, denoted by \(c\), plays a crucial role.Calculate \(c\) using the formula:\[ c^2 = b^2 - a^2 \],where \(b^2\) and \(a^2\) are the squares of the lengths of the semi-major and semi-minor axes, respectively. For this ellipse, the values are \(b^2 = 5\) and \(a^2 = 2\):

• \(c^2 = 5 - 2 = 3\)
• \(c = \sqrt{3} \approx 1.73\)

Thus, since the ellipse's major axis is vertical, the foci are located at \((0, \pm 1.73)\). These points, just like the vertices, are aligned along the y-axis due to the vertical structure of the ellipse.

The **standard form** of an ellipse is essential as it reveals many properties of the ellipse at a glance. This form is represented mathematically as:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] or \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \], depending on whether the ellipse is horizontal or vertical.For the given example, simplifying the initial equation led us to the standard form:\[ \frac{x^2}{2} + \frac{y^2}{5} = 1 \].

• Here, \(a^2 = 2\) and \(b^2 = 5\).
• Since \(b^2 > a^2\), it confirms a vertical ellipse with the major axis along the y-axis.

This daunting equation simplifies understanding and calculating the ellipse's diverse elements, such as axes, vertices, and foci.

For any ellipse, the **semi-major axis** is the longer radius extending from the center of the ellipse to the edge, along the longest line that passes through the ellipse. It plays a vital role in defining the ellipse's size and orientation.In our problem:

• Since \(b^2 = 5 > 2 = a^2\), the semi-major axis is associated with the \(y\) term.
• Thus, the length of the semi-major axis is \(b = \sqrt{5} \approx 2.24\).

This is the farthest distance from the center \((0,0)\) of the ellipse to the vertices along the vertical line, making it crucial for understanding the overall height of the ellipse on a graph.

The **semi-minor axis** is defined as the shortest radius from the ellipse's center to its perimeter. It is shorter than the semi-major axis and, combined with the semi-major axis, defines the ellipse's "width."Following the analysis of our equation:

• Given \(a^2 = 2 < b^2 = 5\), the semi-minor axis corresponds to the \(x\) direction.
• The length of the semi-minor axis is \(a = \sqrt{2} \approx 1.41\).

The semi-minor axis helps determine the width and overall shape of the ellipse, influencing how stretched or compact it appears horizontally on a graph.