Problem 15 - A Graphical Approach to College Algebra

Question

Graph each ellipse by hand. Give the domain and range. Give the foci and identify the center. Do not use a calculator. $$\frac{25 y^{2}}{36}+\frac{64 x^{2}}{9}=1$$

Step-by-Step Solution

Answer

Center: (0,0), Domain: [-0.375,0.375], Range: [-1.2,1.2], Foci: (0,±1.14).

Step 1: Identify the Form of the Ellipse

The given equation is \(\frac{25 y^{2}}{36} + \frac{64 x^{2}}{9} = 1\). It can be rearranged to the standard form of an ellipse \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), where \(a^2 = 36/25\) and \(b^2 = 9/64\). This equation indicates a vertical ellipse because \(\frac{25 y^2}{36}\) suggests that the term with \(y^2\) is larger, making the terms ‘\(a^2\)’ and ‘\(b^2\)’ switched in their positions.

Step 2: Determine the Center of the Ellipse

The center of the ellipse is derived from the equation \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\). Since there are no shifts in \(x\) or \(y\), \(h = 0\) and \(k = 0\), placing the center of the ellipse at \((0, 0)\).

Step 3: Find the Lengths of the Major and Minor Axes

For \(y^2\), \(a^2 = 36/25 = 1.44\), so \(a = \sqrt{1.44} = 1.2\). For \(x^2\), \(b^2 = 9/64 = 0.140625\), so \(b = \sqrt{0.140625} = 0.375\). Thus, the major axis length is \(2a = 2(1.2) = 2.4\) and the minor axis length is \(2b = 2(0.375) = 0.75\).

Step 4: Calculate the Foci of the Ellipse

The foci \((0, \pm c)\) are determined using the formula \(c = \sqrt{a^2 - b^2}\). Substitute and calculate \(c = \sqrt{1.44 - 0.140625} = \sqrt{1.299375} \approx 1.14\). Thus, the foci are at \((0, \pm 1.14)\).

Step 5: Determine the Domain and Range

The domain of the ellipse is based on the major axis and is given by \([-0.375, 0.375]\). The range is based on the minor axis and is \([-1.2, 1.2]\).

Key Concepts

Vertical Ellipse

An ellipse can be classified as either horizontal or vertical, based on the orientation of its major axis. In this particular example, we have a vertical ellipse. The standard form of a vertical ellipse's equation is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). Here, the term with \(y^2\) is larger, which indicates that the major axis is vertical. Consequently, the longer dimension of the ellipse is along the y-axis.
When working with a vertical ellipse:

• The major axis is aligned vertically.
• The length of the major axis is \(2a\), where \(a\) is the square root of the denominator under \(y^2\).
• The minor axis is horizontal, with length \(2b\).

Understanding these characteristics is crucial for graphing and identifying features of a vertical ellipse.

Center of Ellipse

The center of an ellipse is where its two axes intersect. In standard form, the equation of an ellipse is \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\), where \(h\) and \(k\) are the coordinates of the center. For the provided equation, \(\frac{25 y^{2}}{36} + \frac{64 x^{2}}{9}=1\), there are no horizontal or vertical shifts applied to \(x\) or \(y\). Thus, this tells us directly that the center of the ellipse is at \((0, 0)\).
The center is crucial for graphing the ellipse correctly, acting as the reference point for plotting axes, foci, and the ellipse itself.

Foci of Ellipse

The foci of an ellipse are two points located along its major axis. They play a key role in its geometric definition, where any point on the ellipse maintains a constant total distance to these foci. For a vertical ellipse like our current one, the foci have coordinates \((0, \pm c)\).
To find \(c\), use the formula \(c = \sqrt{a^2 - b^2}\).

• Here, \(a^2 = 1.44\) and \(b^2 = 0.140625\).
• Therefore, \(c = \sqrt{1.44 - 0.140625} = \sqrt{1.299375} \approx 1.14\).

This gives the foci at approximately \((0, \pm 1.14)\). Knowing the foci assists not only in understanding the ellipse's shape but also its dynamic properties as a geometric figure.

Domain and Range

When discussing an ellipse, the domain and range inform about its extent on the x and y axes, respectively. For the given vertical ellipse, these intervals indicate how far the ellipse stretches horizontally and vertically.
The domain refers to all possible x-values:

• Calculate using the half-length of the minor axis, \(b = 0.375\).
• So the domain is \([-0.375, 0.375]\).

The range refers to all possible y-values:

• It's determined by the half-length of the major axis, \(a = 1.2\).
• Thus, the range becomes \([-1.2, 1.2]\).

Understanding the domain and range helps visualize where the entirety of the ellipse lies on the coordinate plane. These values are essential for sketching the ellipse's graph accurately.