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 in Cartesian coordinates. $$\begin{array}{l}{\text { Vertices: }(0, \pm 70)} \\ {\text { Eccentricity: } 0.1}\end{array}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Type of Ellipse

Given vertices \((0, \pm 70)\), the major axis is along the y-axis because the vertices are vertically aligned with the center \((0,0)\). Thus, the equation of the ellipse will be of the form \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\), where \(a\) is the semi-major axis length.

Step 2: Use the Vertex Information

Since the vertices are \((0, \pm 70)\), the length of the semi-major axis \(a\) is 70. Therefore, \(a^2 = 70^2 = 4900\).

Step 3: Calculate the Foci using Eccentricity

The eccentricity \(e = 0.1\). The formula for eccentricity is \(e = \frac{c}{a}\), where \(c\) is the distance from the center to a focus. So, \(c = ea = 0.1 \times 70 = 7\).

Step 4: Calculate the Semi-minor Axis

The relationship between \(a\), \(b\) (semi-minor axis), and \(c\) is given by \(c^2 = a^2 - b^2\). So, \(7^2 = 4900 - b^2\). Solving for \(b^2\), we get \(b^2 = 4900 - 49 = 4851\).

Step 5: Write the Ellipse Equation

Substituting \(a^2 = 4900\) and \(b^2 = 4851\) into the standard form of the ellipse equation, we get:\[ \frac{x^2}{4851} + \frac{y^2}{4900} = 1 \].

Key Concepts

Cartesian coordinates

Cartesian coordinates are a system that allows us to determine positions on a plane using two numbers, typically referred to as the **x-coordinate** and the **y-coordinate**. Each point on this plane is defined by its distance from the two intersecting perpendicular lines called axes, these are the x-axis (horizontal) and y-axis (vertical).

In the context of ellipses, we use Cartesian coordinates to express the standard form of an ellipse's equation. This form is \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \] where

• **a** represents the length of the semi-major axis,
• **b** represents the length of the semi-minor axis,
• **x** and **y** are any point on the ellipse in the Cartesian plane.

The center of the ellipse is typically at the origin (0,0) in coordinate terms, meaning that it is symmetric around both axes.

eccentricity

Eccentricity is a measure that describes how much an ellipse differs from being a perfect circle. It is a crucial concept as it defines the shape of an ellipse.

In mathematical terms, eccentricity, denoted with **e**, is calculated as:\[ e = \frac{c}{a} \]where:

• **c** is the distance from the center of the ellipse to each focus,
• **a** is the length of the semi-major axis.

Eccentricity values range from 0 to less than 1:

• If **e = 0**, the ellipse is a perfect circle.
• As **e** approaches 1, the ellipse becomes more elongated.

In the exercise, **e** is given as 0.1, indicating that the ellipse is almost circular, but not quite. The low eccentricity value implies a slightly stretched circle along the axis with the semi-major length.

semi-major axis

The semi-major axis of an ellipse is the longest radius from the center of the ellipse to its perimeter. It plays a critical role in determining the overall size and shape of the ellipse.For the ellipse equations, using the standard form \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \],• **a** represents the semi-major axis length. The semi-major axis extends from the center of the ellipse through its widest part, and in our exercise, this length is along the y-axis. The semi-major axis is given as 70, so:

• **a = 70**
• **a^2 = 4900**

This large value of the semi-major axis indicates that the ellipse is elongated in the y-direction, differentiating it from a simple circle.

semi-minor axis

The semi-minor axis is the shortest radius from the center to the ellipse's edge, contrasting the semi-major axis which is the longest.Still using the ellipse equation \[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \],\[ b \] is the semi-minor axis and it dictates how much the ellipse is squished vertically relative to the semi-major axis. To find **b**, we use the relationship between **a**, **b**, and **c**:\[ c^2 = a^2 - b^2 \],where **c** is from the ellipse's eccentricity, calculated as **7** from our given data.

Substituting the numbers, we solved for:\[ b^2 = 4900 - 49 = 4851 \].Using this, we see that **b** is slightly smaller than **a**, giving the ellipse its oval shape, stretched along the length of the semi-major axis.