Problem 61 - A Graphical Approach to Precalculus with Limits

Question

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=\frac{2}{1-\cos \theta}$$

Step-by-Step Solution

Answer

The equivalent rectangular equation is \(y^2 = 4(x+1)\).

Step 1: Understand the Polar Equation

The given equation is in polar form: \( r = \frac{2}{1-\cos\theta} \). Our task is to convert this into rectangular coordinates, which involve \(x\) and \(y\).

Step 2: Convert Polar to Rectangular Coordinates

In polar coordinates, \(x = r\cos\theta\) and \(y = r\sin\theta\). We begin by multiplying both sides of the equation \(r = \frac{2}{1-\cos\theta}\) by \(1-\cos\theta\) to solve for \(r(1-\cos\theta) = 2\).

Step 3: Expand the Equation

The equation \(r(1-\cos\theta) = 2\) expands to \(r - r\cos\theta = 2\). Substituting \(r\cos\theta = x\), we get \(r = x + 2\).

Step 4: Relate r to Squared Terms

We know \(r^2 = x^2 + y^2\). Substitute \(r = x + 2\) into \(r^2 = x^2 + y^2\) to get \((x + 2)^2 = x^2 + y^2\).

Step 5: Solve for the Equivalent Rectangular Equation

Expand \((x + 2)^2\) to get \(x^2 + 4x + 4 = x^2 + y^2\). Simplify to obtain \(4x + 4 = y^2\). Thus, the equivalent rectangle equation is \(y^2 = 4x + 4\).

Step 6: Graph the Rectangular Equation

The equation \(y^2 = 4(x + 1)\) represents a parabola that opens to the right, centered at (-1,0). Draw the graph to visualize this parabola.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that represent points in the polar coordinate system. This system uses a radius and angle to specify the location of a point. In this system:

• The length from the origin to the point is denoted by \(r\), the radius.
• The angle from the positive x-axis to the radius line is marked as \(\theta\), the angle.

These equations are defined using functions of \(r\) and \(\theta\). They provide an alternative method to Cartesian (rectangular) coordinates and are particularly handy in dealing with circular and spiral shapes due to the nature of their formulation.

Graphing Parabolas

Graphing parabolas is an essential part of studying quadratic equations. In the standard rectangular coordinate system, a vertical parabola is often given by an equation like \(y = ax^2 + bx + c\), but in this exercise, we encounter a horizontal parabola in the form \(y^2 = 4(x + 1)\).
A parabola is a curve where the distance to a fixed point, the focus, is equal to the distance to a line, the directrix. For the horizontal parabola \(y^2 = 4(x + 1)\):

• It opens to the right.
• The vertex is located at \((-1, 0)\).

To graph parabolas, identify the vertex, focus, and understand the direction in which the parabola opens. This provides a clear path to graph the shaped curve effectively.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the standard way of plotting points on a grid using horizontal and vertical axes.

• The x-coordinate indicates horizontal position: right (positive) or left (negative) relative to the origin.
• The y-coordinate shows vertical position: up (positive) or down (negative) from the origin.

This system is widely used due to its ease of understanding and is the default choice in most algebraic contexts. Rectangular equations involve expressions with \(x\) and \(y\) and help describe a wide range of mathematical and real-world phenomena.

Coordinate Conversion

Coordinate conversion refers to the process of transforming one form of coordinates to another, most commonly between polar and rectangular coordinates. Understanding the conversion formulas is key:

• The rectangular coordinate \(x\) can be found through the formula \(x = r\cos\theta\).
• The rectangular coordinate \(y\) is obtained via \(y = r\sin\theta\).
• To convert from rectangular back to polar, use \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\frac{y}{x}\).

These conversions are essential for solving problems where a solution is needed in a different coordinate system than the one initially provided, ensuring a comprehensive understanding across mathematical problems and solutions.