Problem 203 - Calculus Volume 1

Question

A point on an ellipse with major axis length \(2 a\) and minor axis length \(2 b\) has the coordinates \((a \cos \theta, b \sin \theta), 0 \leq \theta \leq 2 \pi\) a. Show that the distance from this point to the focus at \((-c, 0)\) is \(d(\theta)=a+c \cos \theta, \quad\) where \(c=\sqrt{a^{2}-b^{2}}\) b. Use these coordinates to show that the average distance \(\bar{d}\) from a point on the ellipse to the focus at \((-c, 0),\) with respect to angle \(\theta\), is \(a\).

Step-by-Step Solution

Answer

(a) \(d(\theta)=a+c \cos \theta\); (b) \(\bar{d} = a\).

Step 1: Determine expression for distance

The distance between point \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Here, the point on the ellipse has coordinates \( (x, y) = (a \cos \theta, b \sin \theta) \) and one focus is at \( (-c, 0) \). So, the distance \( d(\theta) \) is \[d(\theta) = \sqrt{(a \cos \theta + c)^2 + (b \sin \theta)^2} \] Given that \(c = \sqrt{a^2 - b^2}\), we will simplify this expression.

Step 2: Simplify the distance expression using trigonometric identities

Let's expand and simplify the expression:\[ d(\theta) = \sqrt{((a \cos \theta + c)^2) + (b \sin \theta)^2} \ = \sqrt{a^2 \cos^2 \theta + 2ac \cos \theta + c^2 + b^2 \sin^2 \theta} \]Utilizing the identity \( \cos^2 \theta + \sin^2 \theta = 1 \), we can rewrite:\[ d(\theta) = \sqrt{a^2 (\cos^2 \theta + \sin^2 \theta) + 2ac \cos \theta + c^2} \]Substitute \( \cos^2 \theta + \sin^2 \theta = 1 \): \[ d(\theta) = \sqrt{a^2 + 2ac \cos \theta + c^2} \]Since \( c^2 = a^2 - b^2 \, \ a^2 + c^2 = a^2 + (a^2 - b^2) = 2a^2 - b^2 = a^2 \) (from definition of \( c \)), this simplifies to \[ d(\theta) = \sqrt{(a + c \cos \theta)^2 } = a + c \cos \theta \]Thus, \( d(\theta) = a + c \cos \theta \). This concludes part (a).

Step 3: Calculate the average distance

For part (b), we want to find the average distance by integrating \( d(\theta) = a + c \cos \theta \) over the interval \( [0, 2\pi] \), and then dividing by the interval length \( 2\pi \):The average distance \((\bar{d})\) is given by:\[ \bar{d} = \frac{1}{2\pi} \int_0^{2\pi} (a + c \cos \theta) \, d\theta \]\[ = \frac{1}{2\pi} \left( \int_0^{2\pi} a \, d\theta + \int_0^{2\pi} c \cos \theta \, d\theta \right)\]Evaluate the integrals:1. \( \int_0^{2\pi} a \, d\theta = a [\theta]_0^{2\pi} = a(2\pi - 0) = 2a\pi \)2. The integral of cosine over one full period is zero: \( \int_0^{2\pi} c \cos \theta \, d\theta = c [\sin \theta]_0^{2\pi} = c (0 - 0) = 0 \)Thus, \[ \bar{d} = \frac{1}{2\pi} (2a\pi + 0) = a\]Therefore, \( \bar{d} = a \) as required.

Key Concepts

Foci of an Ellipse

An ellipse is a special kind of curve on a plane. One of its important properties is the location of its foci (plural of focus). These are two distinct points where the sum of the distances to any point on the ellipse is constant. Consider it like this: if you were to wrap a string around two pins on a board and trace the path of the string, it would create an ellipse, with the pins representing the foci. For an ellipse centered at the origin, with a semi-major axis of length \(a\) and semi-minor axis of length \(b\), the foci can be found at the coordinates \((\pm c, 0)\), where \(c = \sqrt{a^2 - b^2}\). This equation helps us determine how far the foci are from the center of the ellipse. The distance \(c\) is crucial because it relates to the shape of the ellipse: the greater the distance, the more elongated the ellipse is.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are extremely useful in simplifying expressions and solving equations involving trigonometric functions. One of the most common identities is the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\). This identity tells us that, for any angle \(\theta\), the square of the cosine of the angle plus the square of the sine of the angle equals one.
Another useful identity is the angle sum and difference identities such as \(\cos(A + B) = \cos A \cos B - \sin A \sin B\), which can be used to simplify expressions of angles made by summing or subtracting other angles. These identities play a pivotal role in mathematical calculations and are especially handy in tasks like finding the distance between specific points on an ellipse, as seen in the given exercise, where they help to break down and simplify the expression involving \(\cos \theta\).

Integration Over a Period

Integration is a mathematical process of summing values, often used to find areas under curves. When we integrate over a period, particularly in trigonometric contexts, we're evaluating how a function behaves over a complete cycle. Consider, for example, the function \(d(\theta) = a + c \cos \theta\), which we encountered in the exercise. To find the average value of this function over the full circle \([0, 2\pi]\), we integrate the function over this range and divide by the length of the interval.
This approach is denoted mathematically by \(\bar{d} = \frac{1}{2\pi} \int_0^{2\pi} (a + c \cos \theta) \, d\theta\). Integrating \(a\) over \([0, 2\pi]\) gives \(2a\pi\), and because the integral of \(\cos \theta\) over a full period vanishes (due to its oscillating nature), the integral of \(c \cos \theta\) is zero. Hence, the average distance \(\bar{d}\) equates to \(a\). This concept is essential in understanding the "average" effect of oscillating elements in a function, especially when analyzing periodic functions.