Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-30: Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. Vertices \(V(0, \pm 7)\), foci \(F(0, \pm 2)\)

Step-by-Step Solution

Answer

\(\frac{x^2}{45} + \frac{y^2}{49} = 1\)

Step 1: Identify the form of the ellipse equation

The general equation of an ellipse centered at the origin with a vertical major axis is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), where \(a > b\). The vertices and foci help us find values for \(a\), \(b\), and \(c\).

Step 2: Determine \(a\) from the vertices

The distance from the center to each vertex along the major axis is \(a\). Since the vertices are \(V(0, \pm 7)\), we have \(a = 7\).

Step 3: Determine \(c\) from the foci

The distance from the center to each focus along the major axis is \(c\). The foci are given as \(F(0, \pm 2)\), which gives \(c = 2\).

Step 4: Use the relationship \(c^2 = a^2 - b^2\)

We know \(a^2 = 49\) and \(c^2 = 4\). Substitute these values into the equation \(c^2 = a^2 - b^2\) to find \(b^2\): \[4 = 49 - b^2\].

Step 5: Solve for \(b^2\)

Rearrange the equation to solve for \(b^2\): \[b^2 = 49 - 4 = 45\]. This gives us \(b^2 = 45\).

Step 6: Plug \(a^2\) and \(b^2\) into the ellipse equation

Now we have \(b^2 = 45\) and \(a^2 = 49\). Substitute these into the standard form equation: \[\frac{x^2}{45} + \frac{y^2}{49} = 1\]. This is the equation of the ellipse.

Key Concepts

Vertices of Ellipse

To understand the vertices of an ellipse, it's essential to know their significance. Vertices are the points where the ellipse is widest along its major axis. Imagine the long axis of an ellipse. The tips of this axis are known as the vertices. For our given problem, we have vertices at the coordinates \(V(0, \pm 7)\). These coordinates tell us that the ellipse extends 7 units up and down from its center. The major axis in this case is vertical because the vertices are located along the y-axis.
The distance from the center to a vertex gives the value of \(a\), which represents the semi-major axis length. In this problem, \(a = 7\). Remember this value because it’s fundamental to writing the equation of our ellipse.
In the general ellipse equation form \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), \(a\) is particularly crucial as it dictates the proportions and direction of the ellipse along its major axis.

Foci of Ellipse

The foci (pronounced FO-sigh) of an ellipse are two special points located on the major axis inside the ellipse. These points have a unique property that defines the shape of the ellipse: the sum of the distances from any point on the ellipse to the two foci is always constant.
In our exercise, the foci are given as \(F(0, \pm 2)\), indicating they are 2 units above and below the center. This placement of the foci confirms that the major axis is vertical, aligning with our vertices.
To find these foci, you often use the relation \(c^2 = a^2 - b^2\), where \(c\) is the distance from the center to a focus. This formula arises from the Pythagorean relationship inherent in ellipse geometry. With \(c = 2\) for our ellipse, we've identified a key property that helps us determine the ellipse's characteristics.

Center of Ellipse

Every ellipse has a central point, which is the "balance point" of its symmetrical shape. For ellipses centered at the origin, this point is \((0,0)\). The center is crucial as it serves as the reference point for measuring distance to the vertices and foci.
In our problem, the ellipse's center is explicitly at the origin, \((0, 0)\). This means that both the vertices and foci are symmetrically distributed around the y-axis, perfectly aligned vertically.
The center aids significantly in forming the ellipse equation because it simplifies formulas to their origin-centered form like \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\). Knowing the center's location helps in setting up the coordinate-based calculations for distance and allows for straightforward plotting.

Major and Minor Axes

The ellipse features two primary axes: the major and minor axes. The major axis is the longest diameter of the ellipse, running through the two vertices. In our case, it is the y-axis. The minor axis is shorter, perpendicular to the major, and runs through the center of the ellipse.
For an ellipse centered at the origin, the major axis is aligned along one of the coordinate axes. Given our vertices, the major axis length is twice \(a\), here \(14\), because \(a = 7\). The minor axis, determined by \(b\), has a total length of \(2\sqrt{45}\), emphasizing how \(b\)'s value was deduced from the relation \(c^2 = a^2 - b^2\).
Understanding the axes helps in visualizing the ellipse's shape and aids in drawing or graphing it, as you can see the stretching or squeezing depending on the comparison of \(a\) and \(b\). Each axis length directly influences the boundary that encapsulates all the points of the ellipse.