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 \((-c, 0) \quad\) is \(\quad d(\theta)=a+c \cos \theta, \quad\) where \(c=\sqrt{a^{2}-b^{2}}\) b. Use these coordinates to show that the average distance \(\overline{d}\) from a point on the ellipse to the focus at \((-c, 0),\) with respect to angle \(\theta,\) is \(a .\)

Step-by-Step Solution

Answer

The distance is \(d(\theta)=a+c\cos\theta\) and the average distance is \(a\).

Step 1: Identify Coordinates and Focus

The point on the ellipse is given by \( (a \cos \theta, b \sin \theta) \). The focus of the ellipse is at \((-c, 0)\) where \(c=\sqrt{a^2-b^2}\).

Step 2: Distance Formula Setup

To find the distance \(d(\theta)\), we use the distance formula between two points: \[ d(\theta) = \sqrt{((a \cos \theta) - (-c))^2 + ((b \sin \theta) - 0)^2} \] This simplifies to:\[ d(\theta) = \sqrt{(a \cos \theta + c)^2 + (b \sin \theta)^2} \]

Step 3: Simplify the Expression

Expand and simplify the inside of the square root:\[ d(\theta) = \sqrt{a^2 \cos^2 \theta + 2ac \cos \theta + c^2 + b^2 \sin^2 \theta} \]Use the identity \(a^2 \cos^2 \theta + b^2 \sin^2 \theta = a^2 - (a^2-b^2)\sin^2 \theta = a^2 - c^2 \sin^2 \theta\):\[ d(\theta) = \sqrt{a^2 - c^2 \sin^2 \theta + 2ac \cos \theta + c^2} \]

Step 4: Apply Specific Trigonometric Identity

Recognize that \(c^2 \sin^2 \theta\) and \(c^2\) cancel partially:\[ d(\theta) = \sqrt{a^2 + 2ac \cos \theta} \]Factor:\[ d(\theta) = a+c \cos \theta \] by recognizing that the square root simplifies due to cancellation of terms.

Step 5: Integrate to Find Average Distance

To find the average distance \(\overline{d}\), integrate \(d(\theta)\) over one period \([0, 2\pi]\) and divide by the total angle:Integration: \[ \int_0^{2\pi} (a + c \cos \theta) \, d\theta = \int_0^{2\pi} a \, d\theta + \int_0^{2\pi} c \cos \theta \, d\theta \]The first integral: \[ a \cdot 2\pi \]The second integral vanishes: \[ \left[ c \sin \theta \right]_0^{2\pi} = 0 \]Average: \[ \overline{d} = \frac{a \cdot 2\pi}{2\pi} = a \].

Key Concepts

distance formula

The distance formula is a crucial tool for measuring the distance between two points in a coordinate plane. In general, the formula to calculate the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. This formula helps determine the length of the direct line connecting both points.
In the context of an ellipse, it helps us find the distance from a point on the ellipse to a focus. Using this formula, and given a point on the ellipse \((a \cos \theta, b \sin \theta)\) and a focus at \((-c, 0)\), we calculate the distance \(d(\theta)\) as: \[ d(\theta) = \sqrt{((a \cos \theta) - (-c))^2 + ((b \sin \theta) - 0)^2} \]. This simplifies further to a more manageable form using algebra and trigonometric identities.

trigonometric identities

Trigonometric identities are formulas that allow us to simplify expressions involving trigonometric functions like sine and cosine. They're vital for manipulating and solving trigonometric equations. Some key trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Double Angle: \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

In problems involving ellipses, we can use these identities to simplify expressions. For instance, rewriting the term \(a^2 \cos^2 \theta + b^2 \sin^2 \theta\):
By recognizing \( a^2 \cos^2 \theta + b^2 \sin^2 \theta = a^2 - (a^2 - b^2) \sin^2 \theta \) and using \(c = \sqrt{a^2 - b^2}\), we can convert this into a simpler form. Trigonometric identities streamline calculations, making them more straightforward and less error-prone.

average distance

The average distance between a point on an ellipse and a focus can be calculated by integrating the distance formula over the complete angle \(\theta\) and dividing by the entire angular range. This process gives us an understanding of the mean separation over one full rotation.
For example, given that \(d(\theta) = a + c \cos \theta\), we integrate over the interval \([0, 2\pi]\): \[ \int_0^{2\pi} (a + c \cos \theta) \, d\theta \]. This results in two separate integrals: one for \(a\) and one for \(c \cos \theta\). The integral \(\int_0^{2\pi} a \, d\theta\) evaluates to \(a \times 2\pi\), while \(\int_0^{2\pi} c \cos \theta \, d\theta\) evaluates to zero. Hence, the average distance \(\overline{d}\) simplifies to \(a\).
Integration and averaging provide a clearer picture of how the distance behaves across the ellipse, highlighting interesting geometrical properties.

focus of an ellipse

The focus of an ellipse is a critical point. It represents one of two fixed points from which the sum of the distances to any point on the ellipse is constant. An ellipse has two foci. Their positioning is determined by the relation \(c = \sqrt{a^2 - b^2}\), where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. The foci lie on the major axis at points \((-c, 0)\) and \((c, 0)\).
The focus plays a significant role in defining the shape and orientation of the ellipse. In an exercise where we calculate the distance from a point on the ellipse to the focus, understanding its location is key. The interplay between the distances from any point on the ellipse to the foci underscores many of the unique properties of ellipses, such as reflection properties and optimization problems.