Problem 27 - Algebra and Trigonometry

Question

In Exercises \(25-36,\) find the standard form of the equation of each ellipse satisfying the given conditions. Foci \((0,-4),(0,4) ;\) vertices: \((0,-7),(0,7)\)

Step-by-Step Solution

Answer

The standard form equation of the ellipse is \( \frac{x^2}{33} + \frac{y^2}{49} = 1 \).

Step 1

Identify the values of 'a' and 'c'. The distance from the center to a vertex (a value) is 7 and the distance from the center to a focus (c value) is 4.

Step 2

Calculate the value of 'b' using the equation \( a^2 = b^2 + c^2 \). This can be rearranged to find 'b' as \( b = \sqrt{a^2 - c^2} = \sqrt{49 - 16} = \sqrt{33} \).

Step 3

Use the values for 'a' and 'b' to write the standard form equation of the ellipse. Since the major axis is vertical, the larger denominator is under the y-term: \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \). Substitute 'a' = 7 and 'b' = \sqrt{33} to get \( \frac{x^2}{33} + \frac{y^2}{49} = 1 \).

Key Concepts

Standard Form of an Ellipse

The standard form of an ellipse is an equation that provides a way to describe its shape and position on a coordinate plane. This form is particularly useful because it allows us to quickly determine the major and minor axes of the ellipse, as well as its orientation.

An ellipse's standard form will differ based on whether it's horizontally or vertically oriented. If the major axis is vertical, the standard form is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \), where \(a\) is the distance from the center to the vertices along the y-axis, and \(b\) is the distance from the center to the co-vertices along the x-axis. Conversely, if the major axis is horizontal, the equation flips, with \(a\) associated with the x-axis.

In the exercise provided, we have an ellipse with a vertical major axis since the vertices and foci have the same x-coordinate but different y-coordinates. Therefore, using the exercise's found values of \(a = 7\) and \(b = \sqrt{33}\), the standard form of that particular ellipse is \( \frac{x^2}{33} + \frac{y^2}{49} = 1 \). This equation encapsulates the shape of an upright elongated circle—our ellipse.

Foci of an Ellipse

The foci of an ellipse are two fixed points located along the major axis that are of paramount importance in defining the properties of an ellipse. For any point on the ellipse, the sum of the distances to the foci is constant, and this trait helps to maintain the ellipse's unique shape.

The distance between the center of the ellipse and each focus is known as the \(c\) value. The relationship between \(a\), \(b\), and \(c\) in an ellipse is given by the equation \(a^2 = b^2 + c^2\), where \(a\) is the semi-major axis, \(b\) the semi-minor axis, and \(c\) the focal length. Knowing the location of the foci allows us to understand the shape of the ellipse better. In the given problem, the foci are located at (0,-4) and (0, 4), which suggests a vertical orientation and that \(c = 4\).

This characteristic is essential when drawing an ellipse or analyzing its geometric properties. Ellipses are often utilized in physics and engineering contexts, such as in the orbits of planets and the design of reflective surfaces like satellite dishes, where the foci play a crucial role.

Vertices of an Ellipse

The vertices of an ellipse are the points where the ellipse intersects its major axis, being the furthest points from the center along this crucial axis. There are two vertices, each one at one end of the major axis. Understanding the vertices allows us to determine the length of the major axis and thus the size of the ellipse.

In a standard form equation, the value \(a\) is associated with the vertices; it is the semi-major axis of the ellipse. For the problem at hand, we have the vertices located at (0, -7) and (0, 7), indicating that \(a = 7\). This tells us that the major axis is 14 units long, double the distance from the center to a vertex.

Vertices also help determine if an ellipse is more 'stretched' along the x-axis or y-axis—essentially, they give us the orientation of the ellipse. In this instance, since the vertices have the same x-coordinate but differ in the y-coordinate, we confirm that the major axis is vertical, and thus, the ellipse is taller than it is wide.