Problem 44 - Precalculus Mathematics for Calculus

Question

Find an equation for the ellipse that satisfies the given conditions. Eccentricity: \(\sqrt{3} / 2,\) foci on \(y\) -axis, length of major axis: 4

Step-by-Step Solution

Answer

The equation is: \( x^2 + \frac{y^2}{4} = 1 \).

Step 1: Understanding the Problem

We need to find the equation of an ellipse. The eccentricity is given as \( \varepsilon = \frac{\sqrt{3}}{2} \) and the major axis length is 4. The foci of the ellipse are on the \( y \)-axis, indicating a vertical major axis.

Step 2: Identify the Form of the Ellipse Equation

Since the foci are on the \( y \)-axis, it implies the ellipse has a vertical major axis. The general form of such an ellipse is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) where \( a > b \).

Step 3: Determine Major and Minor Axis Lengths

The major axis length is 4, so \( 2a = 4 \), giving \( a = 2 \). The formula for eccentricity in terms of \( a \) and \( c \) (the distance from the center to a focus) for a vertical ellipse is \( \varepsilon = \frac{c}{a} \).

Step 4: Calculate the Distance to Foci (c)

Use the eccentricity formula \( \varepsilon = \frac{c}{a} \) to find \( c \):\[ \varepsilon = \frac{c}{a} = \frac{\sqrt{3}}{2}. \]Since \( a = 2 \), we get:\[ \frac{c}{2} = \frac{\sqrt{3}}{2} \Rightarrow c = \sqrt{3}. \]

Step 5: Determine b using Relationship of a, b, and c

Recall that for an ellipse with a vertical major axis, \( b^2 = a^2 - c^2 \). Substitute known values:\[ b^2 = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1. \]Thus, \( b = 1. \)

Step 6: Write the Equation of the Ellipse

With \( a \) and \( b \) found, the equation of the ellipse is:\[ \frac{x^2}{1^2} + \frac{y^2}{2^2} = 1 \Rightarrow x^2 + \frac{y^2}{4} = 1. \]

Key Concepts

Eccentricity

Eccentricity of an ellipse is a measure that tells us how much the shape deviates from being a perfect circle. It is denoted by \( \varepsilon \) and calculated as the ratio \( \frac{c}{a} \), where \( c \) is the distance from the center to each focus, and \( a \) is the semi-major axis length. For an ellipse, the eccentricity lies between 0 and 1.

• An eccentricity of 0 means the shape is a perfect circle.
• An eccentricity close to 1 indicates an elongated shape.

In our example, the eccentricity is given as \( \varepsilon = \sqrt{3}/2 \approx 0.866 \), showing that the ellipse is quite elongated.
The eccentricity plays a crucial role in determining the position of the foci. The distance \( c \) can be computed using the equation \( \varepsilon = \frac{c}{a} \). This gives us a sense of how stretched the ellipse is along its major axis.

Vertical Major Axis

An ellipse can have its major axis oriented either horizontally or vertically. A vertical major axis places its longest dimension from top to bottom.
When solving ellipses, knowing whether the major axis is vertical or horizontal determines which variable will have the leading coefficient in its equation.

• A vertical major axis implies the form \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), with \( a > b \).
• The semi-major axis \( a \) affects the \( y \)-term because the primary stretching of the ellipse goes along the y-axis.

In the given exercise, the foci were on the \( y \)-axis, confirming a vertical major axis. With \( a = 2 \), the ellipse extends to 4 units along the \( y \)-axis. This detail is critical when applying the general form equation to ensure the configuration fits with the provided geometrical features.

Ellipse Foci

Foci are special points located inside the ellipse and have a vital role in the definition and construction of the ellipse.
They are located along the major axis, symmetrically around the center.

• In a vertical ellipse, foci are along the \( y \)-axis at points \( (0, c) \) and \( (0, -c) \).
• The distance \( c \), which is the focal distance, is calculated through \( c = a\varepsilon \).

For our ellipse, we have \( c = \sqrt{3} \). This tells us that the foci are located at \( (0, \sqrt{3}) \) and \( (0, -\sqrt{3}) \).
These positions help to further solidify the orientation and shape of the ellipse, verifying its characteristics like the eccentricity and vertical major axis.