Problem 11 - Thomas Calculus

Question

Exercises \(9-12\) give the foci or vertices and the eccentricities of ellipses centered at the origin of the \(x y\) -plane. In each case, find the ellipse's standard-form equation. $$ \begin{array}{l}{\text { Vertices: }(0, \pm 70)} \\ {\text { Eccentricity: } 0.1}\end{array} $$

Step-by-Step Solution

Answer

\(\frac{x^2}{4851} + \frac{y^2}{4900} = 1\)

Step 1: Identify the form of the ellipse

Since the vertices are given as \((0, \pm 70)\), the major axis is vertical. Therefore, the standard form of the ellipse is \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\).

Step 2: Determine the value of 'a'

The vertices give the length of the semi-major axis as \(a = 70\). Thus, \(a^2 = 4900\).

Step 3: Use the eccentricity to find 'c'

The eccentricity \(e\) is defined as \(e = \frac{c}{a}\). Given \(e = 0.1\) and \(a = 70\), we solve for \(c\):\[ c = 0.1 \times 70 = 7. \]

Step 4: Calculate 'b' using the relationship between a, b, and c

Using the equation \(c^2 = a^2 - b^2\), we substitute the known values:\[ 7^2 = 4900 - b^2 \]\[ 49 = 4900 - b^2 \] \[ b^2 = 4900 - 49 = 4851. \]

Step 5: Write the standard form equation of the ellipse

Substitute \(a^2 = 4900\) and \(b^2 = 4851\) into the standard form equation:\[ \frac{x^2}{4851} + \frac{y^2}{4900} = 1. \]

Key Concepts

Standard Form Equation

The standard form equation of an ellipse is fundamental in understanding its geometric properties. For an ellipse oriented with a vertical major axis centered at the origin, the equation is expressed as:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \]
This form is derived based on the orientation of the ellipse's major axis.

• \(a^2\) is the square of the length of the semi-major axis.
• \(b^2\) is the square of the length of the semi-minor axis.

In the given problem, as the vertices are
\((0, \pm 70)\), it indicates the major axis aligns vertically. This positional information allows us to apply the above standard form of the equation to represent the ellipse accurately.

Eccentricity

Eccentricity is a key characteristic that defines the shape of an ellipse. It is denoted by \(e\) and describes how much an ellipse deviates from being a circle.
Eccentricity values range from 0 to 1, where:

• \(e = 0\): Represents a perfect circle.
• \(0 < e < 1\): Indicates an ellipse.

In our exercise, the eccentricity is given as 0.1. This value, being close to zero, implies the ellipse appears nearly circular but is still elongated along its major axis.
The relationship of eccentricity to other properties of the ellipse helps in determining the focus. It is given by the formula:
\[ e = \frac{c}{a} \] where:

• \(c\) is the distance from the center to each focus.
• \(a\) is the semi-major axis length.

Semi-major Axis

The semi-major axis is one of the most important measurements of an ellipse. It refers to half of the longest diameter of the ellipse and is denoted by \(a\).
In this specific problem, the length of the semi-major axis is provided directly from the vertices:

• The vertices are located at \((0, \pm 70)\), indicating \(a = 70\).
• Thus, \(a^2 = 4900\).

This helps in forming the correct standard form equation of the ellipse as well as understanding the overall size of the ellipse.
Identifying \(a\) also plays a crucial role in calculating the focal distance \(c\), through its relationship with eccentricity and the semi-minor axis \(b\).

Vertices

Vertices of an ellipse are the endpoints of the major axis. They signify the extreme points in terms of the ellipse's length and give vital geometric details about its orientation. For an ellipse centered at the origin
and having its major axis vertically aligned, the vertices are given by the points

• \((0, +a)\) and \((0, -a)\).

In the current exercise, the vertices are specified as

• \((0, \pm 70)\).
• These coordinates confirm the semi-major axis length \(a = 70\).

This information not only aids in determining the ellipse's precise orientation but also confirms the major axis, which is fundamental for forming the correct mathematical model of the ellipse.