Problem 71 - Precalculus

Question

Equation Show that the polar equation of the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { is } \quad r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta}$$.

Step-by-Step Solution

Answer

The polar equation of the ellipse is derived as \(r^{2}=\frac{b^{2}}{1 - e^{2}\cos^{2}\theta}\) from its given Cartesian form of \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\).

Step 1: Substitute x and y in terms of r and \(\theta\)

Replace x and y in the ellipse equation with \(r\cos\theta\) and \(r\sin\theta\) respectively. This gives us: \[\frac{(r\cos\theta)^{2}}{a^{2}}+\frac{(r\sin\theta)^{2}}{b^{2}}=1 \]

Step 2: Simplify the obtained expression

After the substitution from Step 1, simplify the equation to form a single fraction which yields us:\[\frac{r^{2}\cos^{2}\theta}{a^{2}}+\frac{r^{2}\sin^{2}\theta}{b^{2}}=1\or\r^{2}=\frac{a^{2}b^{2}}{a^{2}\sin^{2}\theta + b^{2}\cos^{2}\theta}\]

Step 3: Express in terms of eccentricity

Use the eccentricity definition \(e^{2} = 1 - \frac{b^{2}}{a^{2}}\) and rearrange it to form \(\frac{b^{2}}{a^{2}} = 1 - e^{2}\), replace \(\frac{b^{2}}{a^{2}}\) in the equation obtained from step 2, to arrive at the desired polar equation of ellipse:\[\frac{r^{2}(1-e^{2})}{a^{2}\sin^{2}\theta + b^{2}\cos^{2}\theta}=1\or\r^{2}=\frac{b^{2}}{1 - e^{2}\cos^{2}\theta}\]

Key Concepts

Converting Cartesian Coordinates to Polar Coordinates

Understanding the process of converting Cartesian coordinates to polar coordinates is crucial in various areas of mathematics and science, especially when dealing with curves like ellipses. Cartesian coordinates use the familiar (x, y) format to denote the position of a point on a two-dimensional plane. Meanwhile, polar coordinates express this location as (r, \theta), where 'r' represents the radial distance from the origin, and '\theta' is the angle from the positive x-axis.

To convert from Cartesian to polar coordinates, you can use the following equations:

• \r = \( \sqrt{x^2 + y^2} \)
• \theta = \( \atan2(y, x) \)

These formulas arise from the trigonometric relationships in a right triangle, where the radial distance 'r' is the hypotenuse.

In the context of an ellipse with the polar equation \( r^2 = \frac{b^2}{1-e^2\cos^2\theta} \), the conversion involves substituting x with \( r\cos\theta \) and y with \( r\sin\theta \) into the Cartesian form of the ellipse equation. As we work through the algebra, we transition from the rectangular (Cartesian) system to the more convenient polar system for this particular shape.

Ellipse Eccentricity

Ellipse eccentricity is a measure of how much an ellipse deviates from being a circle. It's a dimensionless quantity, typically denoted by 'e'. The eccentricity ranges from 0 to 1 for an ellipse, where 0 is a perfect circle and values closer to 1 indicate a more elongated shape.

The formula for eccentricity is given by:

• \( e = \sqrt{1 - \frac{b^2}{a^2}} \)

Here, 'a' is the semi-major axis, the longest radius of the ellipse, and 'b' is the semi-minor axis, the shortest radius. If 'a' equals 'b', the eccentricity becomes zero, and the ellipse is a circle.

In polar coordinates, the eccentricity is used to express the equation of an ellipse concisely, leading to a form that inherently accounts for the shape's elongation. The aforementioned polar equation of an ellipse \( r^2 = \frac{b^2}{1-e^2\cos^2\theta} \) directly incorporates 'e', allowing us to describe its geometry simply in terms of angular position and eccentricity—a powerful way to analyze elliptical orbits in physics and astronomy.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental technique used to make complex equations more manageable. The process involves combining like terms, reducing fractions, and employing algebraic identities. Simplification is essential when deriving equations, such as converting the Cartesian equation of an ellipse into polar form.

To simplify the expression resulting from the substitution of polar coordinates into the Cartesian equation, one typically combines the r terms and consolidates trigonometric functions. This often involves recognizing that \( \sin^2\theta + \cos^2\theta = 1 \) — a fundamental trigonometric identity.

In our polar equation for the ellipse, after appropriate substitutions and simplifications, one needs to manipulate the terms to isolate \( r^2 \) on one side of the equation, culminating into a compact form that clearly exhibits the relationship between the radial distance and the angle: \( r^2 = \frac{b^2}{1-e^2\cos^2\theta} \).

Simplifying algebraic expressions not only helps in solving equations but also aids in a deeper comprehension of the mathematical structures involved, enabling students to tackle more complex problems with confidence.