Problem 99 - Algebra and Trigonometry

Question

a. Showing all steps, rewrite \(r=\frac{1}{3-3 \cos \theta}\) as \(9 r^{2}=(1+3 r \cos \theta)^{2}\) b. Express \(9 r^{2}=(1+3 r \cos \theta)^{2}\) in rectangular coordinates. Which conic section is represented by the rectangular equation?

Step-by-Step Solution

Answer

The given equation \(r=\frac{1}{3-3 \cos \theta}\) can be manipulated to get \(9 r^{2}=(1+3 r \cos \theta)^{2}\). In rectangular coordinates, it forms a Circle described by the equation \(9*(x^{2} +y^{2}) = (1+3x)^{2}\)

Step 1: Rewriting the equation

To rewrite \(r=\frac{1}{3-3 \cos \theta}\) as \(9 r^{2}=(1+3 r \cos \theta)^{2}\), first multiply both sides of the equation by \(3 - 3\cos\theta\): \(r(3-3 \cos \theta)=1\).Next, distribute \(r\) onto \(3 - 3\cos\theta\), giving: \(3r- 3r cos\theta = 1\).Rearrange it to be:\(3r cos \theta + 3r = 1\).Finally, we need to bring it to the form mentioned in the question, and for that, divide every term by 3, square both sides, multiply both sides by 9 to get:\[9 r^{2}=(1+3 r \cos \theta)^{2}\]

Step 2: Converting to rectangular coordinates

To express the equation in rectangular coordinates, one needs to understand that in rectangular coordinates, \(r = \sqrt{x^{2} + y^{2}}\) and \(\cos \theta = \frac{x}{r}\). By substituting these values in to the equation, when \(r\cos\theta\) is replaced by x and \(r^{2}\) is replaced by \(x^{2}+y^{2}\);it gives us \(9*(x^{2} +y^{2}) = (1+3x)^{2}\)

Step 3: Identifying the conic section

This equation is in the standard form of the equation of a circle, \(x^{2}+y^{2}=r^{2}\) where here the radius r is not constant but depends on the value of x, which means the conic section is a Circle.

Key Concepts

Conic Sections

Conic sections are the curves obtained when a cone is intersected by a plane. In mathematics, these shapes are significant since each represents a set of points satisfying a specific equation. The primary types of conic sections are circles, ellipses, parabolas, and hyperbolas. Each conic section has a unique equation in rectangular coordinates, which is the familiar x-y coordinate system most students learn in algebra.

For example, the equation of a circle in rectangular coordinates is given as \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. In the given exercise, converting the polar coordinate equation to rectangular coordinates leads us to an equation resembling that of a circle. Recognizing the form and characteristics of the conic section involved is critical for solving and understanding problems related to conic sections.

Polar Coordinates

Polar coordinates represent another way to locate points on a plane using a radius and an angle rather than x and y coordinates. In this system, the position of a point is determined by \(r\), the distance from the origin, and \(\theta\), the counter-clockwise angle from the positive x-axis. The beauty of polar coordinates lies in their ability to simplify equations and shapes that are difficult to express in rectangular coordinates.

A polar equation like \(r=\frac{1}{3-3 \cos \theta}\) can often provide insight into the shape's symmetry and properties. In our exercise, we begin with a polar equation that we manipulate algebraically to reveal its underlying structure, ultimately converting it into rectangular coordinates to identify the conic section it represents.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable(s) where both sides of the equality are defined. These identities are useful tools in simplifying expressions and solving equations involving trigonometric functions. Some of the foundational identities are the reciprocal, quotient, and Pythagorean identities.

Within the context of this exercise, the use of the identity \(\cos \theta = \frac{x}{r}\) is a key step in transforming a polar equation into a rectangular form. Trigonometric identities bridge the gap between polar and rectangular coordinate systems, allowing for the conversion between them. By applying these identities, students can often reveal the nature of the conic section described by trigonometric equations in polar coordinates.