Problem 3 - Algebra and Trigonometry

Question

The graph of the equation \(\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1\) with \(a>b>0\) is an ellipse with vertices (_, _) and (_, _) and foci \((0, \pm c),\) where \(c=\) __________ \((0, \pm c),\) where \(c=\) is an ellipse with vertices (_, _) and (_, _) and foci (_,_) and (_, _)

Step-by-Step Solution

Answer

Vertices: (0, -a) and (0, a); Foci: (0, -c) and (0, c), where c = \(\sqrt{a^2 - b^2}\).

Step 1: Identify the Ellipse

The given equation is of the standard form for an ellipse: \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), where \(a > b > 0\). This indicates a vertically oriented ellipse.

Step 2: Determine the Vertices

For a vertically oriented ellipse, the vertices are located at \((0, \pm a)\). This comes from the larger value being under \(y\), signifying vertical orientation.

Step 3: Calculate the Foci

The foci of the ellipse are found using the relation \(c = \sqrt{a^2 - b^2}\). The foci are located at \((0, \pm c)\).

Step 4: Compile the Results

From the calculations, the vertices of the ellipse are \((0, -a)\) and \((0, a)\). The foci are \((0, -c)\) and \((0, c)\), where \(c = \sqrt{a^2 - b^2}\).

Key Concepts

Vertices of an Ellipse

In the world of ellipses, vertices hold crucial importance as they define the extent of the ellipse along its major axis. The major axis of an ellipse is the longest diameter that determines its primary direction. When dealing with a vertically oriented ellipse, like in our original exercise, the vertices are positioned along the vertical line, at

• \((0, -a)\)
• \((0, a)\)

This implies that the ellipse stretches from the point \((0, -a)\) to \((0, a)\) vertically. The essential observation here is that the value of \(a\), which is greater than \(b\), influences this vertical spread. Each vertex is a center of elliptical arc, one at each end of the major axis.

Foci of an Ellipse

Foci are two distinctive points inside an ellipse that help in describing its specific shape. In our example, the foci are placed along the major axis, and they determine how "stretched" or "elongated" the ellipse appears. Calculated using the formula

• \(c = \sqrt{a^2 - b^2}\)

we can understand that \(c\) gives the distance from the center to each focus. This ellipse, being vertically oriented, has its foci at

• \((0, -c)\)
• \((0, c)\)

Understanding the position of the foci helps one gauge the eccentricity, or the degree of "ovality," of the ellipse.

Ellipse Equation

An ellipse's equation is fundamental in defining its geometry. The given equation of our ellipse is

• \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\)

This formulation indicates several things:
- The ellipse is centered at the origin point (0,0).
- It is vertically oriented, as the coefficient of

• \(y^2\)

is greater than that of

• \(x^2\).

- The lengths of semi-major and semi-minor axes are determined by \(a\) and \(b\) respectively. This equation anchors the ellipse to the coordinate grid, detailing its proportional dimensions.

Standard Form of Ellipse

The standard form of an ellipse is a neatly organized equation that lays out its key characteristics. Our standard form, repeatedly referenced, is:

• \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\)

In this equation, \(a\) represents the semi-major axis (length from center to vertex), while \(b\) represents the semi-minor axis. For vertically oriented ellipses, the larger value under

• \(y^2\)

confirms this orientation. By analyzing this standard form, one can instantly see whether the ellipse is elongated horizontally or vertically. Displaying an ellipse in standard form simplifies the process of identifying its properties, easing calculations, and enhancing understanding.