Problem 59 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$$

Step-by-Step Solution

Answer

The polar equation is \( r = \sqrt{\frac{1}{\frac{\cos^{2} \theta}{9} + \frac{\sin^{2} \theta}{4}}} \).

Step 1: Understand the Cartesian Equation

We start with the equation \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \). This is the equation of an ellipse in Cartesian coordinates, where the semi-major axis is 3 (along the x-axis) and the semi-minor axis is 2 (along the y-axis).

Step 2: Convert Cartesian Coordinates to Polar Coordinates

In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute these into the equation \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \).

Step 3: Substitute Values into the Equation

Replace \( x \) and \( y \) in the equation with their polar equivalents: \[\frac{(r \cos \theta)^{2}}{9} + \frac{(r \sin \theta)^{2}}{4} = 1.\] Simplify this to get: \[\frac{r^{2} \cos^{2} \theta}{9} + \frac{r^{2} \sin^{2} \theta}{4} = 1.\]

Step 4: Factor Out \( r^{2} \)

Factor out \( r^{2} \) from the left side of the equation: \[r^{2} \left( \frac{\cos^{2} \theta}{9} + \frac{\sin^{2} \theta}{4} \right) = 1.\]

Step 5: Isolate \( r^{2} \)

Divide both sides by \( \left( \frac{\cos^{2} \theta}{9} + \frac{\sin^{2} \theta}{4} \right) \) to isolate \( r^{2} \): \[r^{2} = \frac{1}{\frac{\cos^{2} \theta}{9} + \frac{\sin^{2} \theta}{4}}.\]

Step 6: Simplify the Polar Equation

Simplify the expression: \[r = \sqrt{\frac{1}{\frac{\cos^{2} \theta}{9} + \frac{\sin^{2} \theta}{4}}}.\] This is the equivalent polar equation.

Key Concepts

Cartesian Coordinates

The concept of Cartesian Coordinates is fundamental in understanding how we represent points in a two-dimensional plane. They are named after the French mathematician René Descartes. Here, every point on the plane is specified by a pair of numerical coordinates.

• The first number, commonly called \(x\), is the position along the horizontal axis.
• The second number, \(y\), represents the position along the vertical axis.

Together, these define the coordinate \((x, y)\). For example, if you have a point \( (3, 2) \), it tells us that the point is located 3 units along the x-axis (horizontal) and 2 units along the y-axis (vertical).
Cartesian coordinates are very intuitive and provide a straightforward way to plot points, analyze shapes, and solve geometry problems.

Ellipse Equation

An ellipse is a smooth, closed curve, which is essentially an elongated circle. The equation of an ellipse in Cartesian coordinates is given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where:

• \(a\) is the length of the semi-major axis (along the x-axis if \(a > b\)).
• \(b\) is the length of the semi-minor axis (along the y-axis if \(a > b\)).

In our original exercise, the ellipse is defined by \( \frac{x^2}{9} + \frac{y^2}{4} = 1 \). Here, \(a = 3\) and \(b = 2\), meaning the ellipse is centered at the origin (0,0) and extends 3 units along the x-axis and 2 units along the y-axis. Ellipses are important in many areas of mathematics, physics, and engineering due to their unique properties and the way they model the presence of two focal points.

Coordinate Conversion

Converting between Cartesian and polar coordinates allows us to leverage different systems for solving problems more easily. In polar coordinates, a point is defined by \((r, \theta)\), where:

• \(r\) is the radius or distance from the origin.
• \(\theta\) is the angle from the positive x-axis.

To convert from Cartesian \((x, y)\) to polar \((r, \theta)\):

• Use \(x = r \cos\theta\).
• Use \(y = r \sin\theta\).
• Find \(r\) as \(\sqrt{x^2 + y^2}\).
• Determine \(\theta\) using \(\tan^{-1}(y/x)\).

In our exercise, the coordinate conversion helps transform the ellipse equation from Cartesian to polar form. This is done by substituting polar equivalents, simplifying, and rearranging terms to get the equation in terms of \(r\) and \(\theta\). Learning these conversions is key to solving complex equations in a variety of mathematical contexts.