Problem 31 - Algebra 2

Question

Write an equation of an ellipse for the given foci and co-vertices. foci \(( \pm 14,0),\) co-vertices \((0, \pm 7)\)

Step-by-Step Solution

Answer

The equation of the ellipse is \(\frac{x^2}{225} + \frac{y^2}{49} = 1\)

Step 1: Identify a, b and c Values

We know that the foci of the ellipse are located at \(\pm 14,0\), thus, c = 14. The co-vertices of the ellipse are located at \(0,\pm 7\), thus, b = 7. As the foci lies on the x-axis, and the foci are further from the center than the co-vertices, the major axis must be the x-axis, hence a = c.

Step 2: Find value of a

We can find the value for a using the relationship \(c = \sqrt{a^2 - b^2}\), so \(a = \sqrt{c^2 + b^2} = \sqrt{14^2 + 7^2} = 15\).

Step 3: Write the equation of ellipse

The equation of an ellipse in standard form with a major axis on the x-axis is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substituting the values a = 15 and b = 7 in the equation, the equation of the ellipse is \(\frac{x^2}{15^2} + \frac{y^2}{7^2} = 1\).

Key Concepts

Understanding Foci of an Ellipse

An ellipse is a fascinating shape in mathematics defined by two special points called "foci" (plural of focus). These points, denoted in this problem as \((\pm 14,0)\), have a unique property: any point on the ellipse has a combined distance to these foci that is constant.
The positions of the foci give a lot of insight into the shape's overall dimensions. They help determine the length of the ellipse along its major axis.

• If the foci are along the x-axis, the major axis is horizontal.
• If the foci are along the y-axis, the major axis is vertical.

This layout helps to determine which axis is longer and hence helps decide the orientation of the ellipse. Here, since the foci are on the x-axis, the major axis is horizontal.

The Role of Co-Vertices in an Ellipse

Co-vertices are points that signify the ends of the minor axis in an ellipse. In this problem, they are given as \((0, \pm 7)\). Co-vertices help define the width of the ellipse.
While the foci provide information about the major axis, the co-vertices give clarity about the minor axis, helping you understand the shorter dimension of the ellipse.

• Co-vertices are always perpendicular to the foci.
• They indicate the length of the ellipse along the minor axis.

For this problem, knowing that the co-vertices lie on the y-axis confirms the vertical direction of the minor axis of this ellipse.

Defining the Major Axis of an Ellipse

The major axis is an essential component of an ellipse, representing the longest diameter and passing through both foci. In this scenario, since the foci \((\pm 14,0)\) lie on the x-axis, the major axis is the horizontal one.
The length of the major axis is determined by the variable \(a\), which is solved through the relationship \(c = \sqrt{a^2 - b^2}\). In this equation:

• \(a\) = length from the center to the end of the ellipse on the major axis.
• \(c\) = distance from the center to a focus.
• \(b\) = length from the center to the end on the minor axis.

By substituting \(c = 14\) and \(b = 7\), we found \(a = 15\).

This means our major axis extends 15 units in each direction from the center, along the x-axis.