Problem 43 - Essential Calculus

Question

Show that the polar equation for \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is \(r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta} \cdot \quad\) Ellipse

Step-by-Step Solution

Answer

After converting the Cartesian to polar coordinates and simplifying, it's confirmed that the polar equation for \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is indeed \(r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta}\).

Step 1: Convert Cartesian to Polar Coordinates

The conversion formulas between Cartesian and polar coordinates are \(x = r \cos \theta\) and \(y = r \sin \theta\). We substitute these into the equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) to get \(\frac{(r \cos \theta)^2}{a^2} + \frac{(r \sin \theta)^2}{b^2} = 1\).

Step 2: Simplify the Equation

Rewrite our obtained equation in a simple form by opening brackets to get \( \frac{r^2 \cos ^{2} \theta}{a^2} + \frac{r^2 \sin ^{2} \theta}{b^2} =1\).

Step 3: Further Simplification

Since \(\sin^2θ + \cos^2θ = 1\), this equation can be written as \(r^2 \left(\frac{\cos ^{2} \theta}{a^2} + \frac{\sin ^{2} \theta}{b^2}\right) =1\). We can solve for \(r^2\) to find \(r^2 = \frac{b^2}{1-e^{2} \cos ^{2} \theta}\) given that \(e^2 = 1 - (b^2/a^2)\).

Key Concepts

Ellipse

An ellipse is a smooth, closed curve that looks like an elongated circle. It is described by the equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) in Cartesian coordinates. The parameters \(a\) and \(b\) are the semi-major and semi-minor axes, respectively. These axes dictate the width and height of the ellipse.

In an ellipse, all points on its boundary are such that the sum of the distances from any point on the curve to two fixed points, called foci, is constant. The eccentricity \(e\) of an ellipse is a measure of how "stretched" it is, calculated as \( e = \sqrt{1-\frac{b^{2}}{a^{2}}} \). A smaller \(e\) indicates a shape more similar to a circle, while a larger \(e\) shows a more elongated form. Understanding these properties helps when converting the ellipse's equation from Cartesian to polar coordinates.

Cartesian Coordinates

Cartesian Coordinates are often used to locate points on a plane using a pair of values, typically \(x\) and \(y\). This coordinate system is intuitive and works well for defining the position of points in a two-dimensional space using perpendicular axes.

In the context of an ellipse, Cartesian coordinates can describe the curve with its standard form equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). Here, each term represents one of the axes of the ellipse. Cartesian coordinates can also be helpful when visualizing an ellipse on a two-dimensional graph, as they relate directly to the geometric dimensions of the shape.

• \(x\) coordinate often refers to the horizontal position.
• \(y\) coordinate indicates the vertical position.

This coordinate system makes it easy to perform algebraic operations that can later be translated into different coordinate systems, such as polar coordinates.

Conversion Formulas

Conversion formulas play a critical role when transforming equations from Cartesian to polar coordinates. This is particularly useful for simplifying complex algebraic equations.

In polar coordinates, any point in the plane is represented by \(r\) (the distance from the origin) and \(\theta\) (the angle from the positive x-axis). To convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\), the following formulas are used:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

In the given exercise, these conversions helped transition the ellipse's Cartesian equation \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) into an equation in terms of \(r\) and \(\theta\). This conversion is crucial for applications in physics and engineering where polar coordinates simplify problem-solving.

Using these conversions, the geometry of the ellipse can be understood in terms of distance and angles from a central point, providing a new viewpoint on the shape's behavior.

Equation Simplification

Equation Simplification is a process of rewriting mathematical expressions in their simplest or most efficient form. This can make solving and understanding problems easier by reducing complexity.

During the step-by-step solution of converting the ellipse's equation from Cartesian to polar format, simplification played a key role. Initially, the conversion gave us \(\frac{(r \cos \theta)^2}{a^2} + \frac{(r \sin \theta)^2}{b^2} = 1\). Through simplification, we combined like terms using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), leading to \(r^2 \left(\frac{\cos^2 \theta}{a^2} + \frac{\sin^2 \theta}{b^2}\right) = 1\).

The final step resulted in the polar equation \(r^2 = \frac{b^2}{1-e^2 \cos^2 \theta}\), which is a more elegant representation of the ellipse. Each simplification step moved us closer to understanding the relationship between an ellipse's form in different coordinate systems, highlighting the elegance of these mathematical transformations.