Problem 29 - Precalculus: Functions and Graphs

Question

Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Eccentricity \(\frac{3}{4}\) vertices \(H 0, \pm 4)\)

Step-by-Step Solution

Answer

The equation is \(\frac{x^2}{7} + \frac{y^2}{16} = 1\).

Step 1: Understanding the Eccentricity

The eccentricity of an ellipse, given by the letter \( e \), is defined by the formula \( e = \frac{c}{a} \), where \( c \) is the distance from the center to the focus, and \( a \) is the semi-major axis length. Here, \( e = \frac{3}{4} \).

Step 2: Identifying Semi-Major Axis

The vertices \((0, \pm 4)\) indicate that the semi-major axis \( a \) is 4 along the \( y \)-axis. Since the eccentricity \( e = \frac{3}{4} \), we know that \( a = 4 \).

Step 3: Calculating the Semi-Minor Axis

Use the equation \( b^2 = a^2 - c^2 \), where \( b \) is the semi-minor axis. First, find \( c \) using \( c = ae = 4 \times \frac{3}{4} = 3 \). Then substitute into the formula: \( b^2 = 16 - 9 = 7 \), implying \( b = \sqrt{7} \).

Step 4: Writing the Equation of the Ellipse

Since the major axis is vertical and centered at the origin, the standard form of the ellipse equation is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \). From previous steps, substitute \( b^2 = 7 \) and \( a^2 = 16 \) into the equation to get \( \frac{x^2}{7} + \frac{y^2}{16} = 1 \).

Key Concepts

Eccentricity

The concept of eccentricity is a key factor in understanding ellipses. In geometry, the eccentricity of an ellipse determines how "stretched" it is compared to a circle. Represented by the symbol \( e \), eccentricity varies between \( 0 \) and \( 1 \). A circle, a special type of ellipse, has an eccentricity of \( 0 \), indicating that both axes are of the same length. For an ellipse:

• \( e = \frac{c}{a} \) where \( c \) is the distance from the center to the foci (plural of focus).
• \( a \) is the semi-major axis, the longest radius of the ellipse.

In our case, the eccentricity is given as \( \frac{3}{4} \). This means our ellipse is quite stretched but not to the maximum extent, as maximum stretch at \( e = 1 \) would result in a parabola. Having eccentricity helps understand the curviness and shape of the ellipse.

Semi-Major Axis

A vital part of an ellipse's geometry is the semi-major axis, denoted as \( a \). This is the longest radius of the ellipse and runs from the center to the furthest point on the edge of the ellipse. Think of it as the long radius of an oval shape.

• The semi-major axis is crucial in determining the overall size of the ellipse.
• The length of the semi-major axis helps in calculating other features, such as the ellipse's area and its eccentricity.

In the exercise, the vertices given are \((0, \pm 4)\), so the semi-major axis length is \( 4 \). Since the vertices lie along the y-axis, it indicates our ellipse is vertically oriented. Thus, \( a = 4 \) and this guides us to configure the ellipse equation properly.

Semi-Minor Axis

Another important element of an ellipse is the semi-minor axis, represented by \( b \). The semi-minor axis is the shorter radius of the ellipse and is perpendicular to the semi-major axis.

• It represents how "narrow" the ellipse is compared to its length.
• It can be calculated using the formula \( b^2 = a^2 - c^2 \).

In our example, we've established the semi-major axis \( a = 4 \) and found \( c = 3 \) using the eccentricity. So, using the formula, we calculate \( b^2 = 16 - 9 = 7 \), making \( b = \sqrt{7} \). The semi-minor axis influences the width of the ellipse, creating that classic oval shape.

Vertices

Vertices are the points where the ellipse intersects its axes and are crucial in defining its size and orientation. For our ellipse, the vertices are located at \((0, \pm 4)\).

• These points lie on the semi-major axis, showing the furthest extent of the ellipse from its center along this direction.
• The number and position of vertices help define both the major and minor axes.

With the vertices at \((0, \pm 4)\), we know the semi-major axis is vertical. This information was instrumental in forming the equation of our ellipse: \( \frac{x^2}{7} + \frac{y^2}{16} = 1 \). The vertices guide the placement and scaling of the ellipse on the coordinate plane.