Problem 11 - College Algebra with Corequisite Support

Question

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. \(\frac{x^{2}}{4}+\frac{y^{2}}{49}=1\)

Step-by-Step Solution

Answer

Endpoints: Major (0, 7), (0, -7); Minor (2, 0), (-2, 0). Foci: (0, 3√5), (0, -3√5).

Step 1: Identify the type of ellipse

The equation \( \frac{x^{2}}{4} + \frac{y^{2}}{49} = 1 \) is in the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( b^2 > a^2 \). Thus, this is a vertically oriented ellipse.

Step 2: Determine the standard form of the equation

The given equation \( \frac{x^{2}}{4} + \frac{y^{2}}{49} = 1 \) is already in the standard form for ellipses \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) with \( a^2 = 4 \) and \( b^2 = 49 \).

Step 3: Calculate the lengths of the semi-major and semi-minor axes

For a vertical ellipse, the semi-major axis length \( b \) is \( \sqrt{49} = 7 \), and the semi-minor axis length \( a \) is \( \sqrt{4} = 2 \).

Step 4: Identify the endpoints of the axes

Since it is a vertical ellipse centered at the origin, the major axis endpoints are at \( (0, 7) \) and \( (0, -7) \). The minor axis endpoints are at \( (2, 0) \) and \( (-2, 0) \).

Step 5: Calculate the coordinates of the foci

The distance of the foci from the center is represented by \( c \), which is calculated using \( c = \sqrt{b^2 - a^2} \). Substitute \( b^2 = 49 \) and \( a^2 = 4 \) to get \( c = \sqrt{49 - 4} = \sqrt{45} = 3\sqrt{5} \). Thus, the foci are at \( (0, 3\sqrt{5}) \) and \( (0, -3\sqrt{5}) \).

Key Concepts

Standard Form of Ellipse

The equation of an ellipse is written in a specific form known as the standard form. It describes the ellipse clearly, distinguishing it from other conic sections like circles or hyperbolas. The standard form of an ellipse with its center at the origin is given by:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) for horizontal ellipses where \( a^2 > b^2 \).
• \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \) for vertical ellipses where \( b^2 > a^2 \).

In these equations, \( a \) represents the semi-major axis and \( b \) represents the semi-minor axis. The lengths of these axes help determine the shape and size of the ellipse. In our exercise, \( b^2 = 49 \) and \( a^2 = 4 \), indicating a vertical ellipse with its equation being \( \frac{x^2}{4} + \frac{y^2}{49} = 1 \).

Major and Minor Axes

The major and minor axes of an ellipse are crucial components in defining its geometry. These axes intersect at the center and are perpendicular to each other. The longer axis is called the major axis, while the shorter one is the minor axis.For a vertically oriented ellipse:

• The semi-major axis length is denoted by \( b \), and its full length is \( 2b \).
• The semi-minor axis length is denoted by \( a \), with a total length of \( 2a \).

In our example, \( b = 7 \) and \( a = 2 \). The major axis vertical endpoints can be located at \( (0, 7) \) and \( (0, -7) \), while the minor axis horizontal endpoints are \( (2, 0) \) and \( (-2, 0) \). Knowing these axes helps visualize the orientation and dimensions of the ellipse.

Endpoints and Foci

The endpoints of the axes and the foci are important in identifying key features of an ellipse. Endpoints mark the extreme points along the ellipse's major and minor axes, showing the maximum and minimum extent of the shape.For a vertically oriented ellipse, as in our case, the endpoints have been found as follows:

• Major axis endpoints: \( (0, 7) \) and \( (0, -7) \)
• Minor axis endpoints: \( (2, 0) \) and \( (-2, 0) \)

The foci are two significant points inside the ellipse along the major axis. They help define the ellipse's properties and keep the sum of the distances from any point on the ellipse to each focus as a constant. The foci are calculated using the formula \( c = \sqrt{b^2 - a^2} \), where \( c \) is the distance from the center to each focus. For our ellipse, \( c = \sqrt{49 - 4} = \sqrt{45} = 3\sqrt{5} \). Hence, the foci are located at the points \( (0, 3\sqrt{5}) \) and \( (0, -3\sqrt{5}) \).