Problem 62 - Precalculus with Limits

Question

Find an equation of the ellipse with vertices \((0, \pm8)\) and eccentricity \(e=\frac{1}{2}\).

Step-by-Step Solution

Answer

The equation of the asked ellipse is \(y^2/64 + x^2/36 = 1\).

Step 1: Determine the value of 'a'

As the vertices of the ellipse are shown to be (0, ±8), the distance from the center to either vertex 'a' is 8.

Step 2: Calculate the value of 'b'

Identifying eccentricity 'e' equals to \(\frac{1}{2}\), we can input it into the formula for eccentricity \(e = \sqrt{1 - (\frac{b^2}{a^2})}\) where, \(a=8\). By rearranging the formula, we have \(b = a \cdot \sqrt{1 - e^2} = 8 \cdot \sqrt{1 - \frac{1}{4}} = 8 \cdot \sqrt{\frac{3}{4}} = 6\).

Step 3: Plug a and b values into the standard form

For a vertically oriented ellipse, the general equation is \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\). Substituting the obtained values in this equation, we get \:(\frac{y^2}{8^2}) + (\frac{x^2}{6^2}) = 1\), hence the equation will be \(y^2/64 + x^2/36 = 1\).

Key Concepts

Understanding Vertices

The vertices of an ellipse are key points that define its size. In this case, the ellipse has vertices at \((0, \pm8)\). This tells us that the ellipse is vertically oriented with its center at the origin \((0,0)\). The distance from the center to each vertex is what we refer to as 'a'. So here, \(a = 8\). This indicates how far the ellipse stretches vertically. The vertices are crucial in shaping the ellipse and play a major role in determining the standard form of its equation.
To summarize, remember:

• The vertices determine the length of the major axis.
• 'a' is the distance from the center to the vertex along the major axis.
• In a vertical ellipse, the vertices are along the y-axis.

Grasping Eccentricity

Eccentricity is a measure of how stretched out an ellipse is. It is symbolized by 'e', and for this ellipse, \(e = \frac{1}{2}\). Eccentricity ranges between 0 and 1, where 0 represents a circle (perfectly round), and values closer to 1 indicate a more elongated shape.
This ellipse's eccentricity tells us it's not too elongated.
We use the formula:\[e = \sqrt{1 - \left(\frac{b^2}{a^2}\right)}\]Here, \(b\) is the distance from the center to the endpoint of the minor axis. Solving for \(b\), with \(a = 8\), we get:

• \(b = 8 \cdot \sqrt{\frac{3}{4}} = 6\)

So, knowing the eccentricity allows us to understand the ellipse's form and calculate 'b'.

The Standard Form of an Ellipse

The standard form of an ellipse differs depending on its orientation. Since this ellipse is vertical, its equation reads:
\[\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\]
By substituting \(a = 8\) and \(b = 6\), we construct its specific equation:
\[\frac{y^2}{64} + \frac{x^2}{36} = 1\]

In this form:

• \(\frac{y^2}{64}\) handles the vertical stretch.
• \(\frac{x^2}{36}\) covers the horizontal span.

This standard form allows quick identification of the ellipse's dimensions and orientation, making it straightforward to graph and analyze.
Memorizing this form is helpful in dealing with similar problems efficiently.