Problem 47 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ x^{2}+y^{2}+x=\sqrt{x^{2}+y^{2}} $$

Step-by-Step Solution

Answer

The polar equation is \( r = 1 - \cos \theta \).

Step 1: Convert Rectangular Equation to Polar Forms

Recall that in polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). Thus, \( x^2 + y^2 = r^2 \). Substitute these into the given equation: \[ x^2 + y^2 + x = \sqrt{x^2 + y^2} \] becomes \[ r^2 + r \cos \theta = r \].

Step 2: Simplify the Polar Equation

Simplify the equation from Step 1: \[ r^2 + r \cos \theta = r \] Divide both sides by \( r \) (assuming \( r eq 0 \)) to obtain \[ r + \cos \theta = 1 \].

Step 3: Rearrange to Get the Polar Equation

Rearrange the simplified equation to express \( r \) explicitly: \[ r = 1 - \cos \theta \]. This is the polar equation corresponding to the given rectangular equation.

Key Concepts

Rectangular to Polar Conversion

When converting from rectangular to polar coordinates, it is essential to understand the relationship between these two systems. Rectangular coordinates, often denoted by \((x, y)\), are based on a Cartesian grid, using horizontal and vertical distances. Polar coordinates, described by \((r, \theta)\), are based on a distance from a reference point and an angle from a reference direction.
To convert rectangular coordinates to polar coordinates, use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)
• \( \theta = \tan^{-1}(y/x) \)

These conversions help in transforming equations into a polar form, which can simplify solving problems by leveraging the radial nature of the polar system. Understanding these relationships makes it easier to switch between coordinate systems for different types of problems.

Polar Equations

Polar equations are mathematical expressions that describe relationships between the radius \( r \) and angle \( \theta \) in a polar coordinate system. Unlike their rectangular counterparts, polar equations neatly represent curves like circles and spirals with potentially simpler forms.
An important aspect of polar equations is their ability to show symmetry and periodicity, which can be challenging to see in rectangular terms. For example, the polar equation \( r = 1 - \cos \theta \) can represent a Limaçon, a type of curve with distinct and interesting properties.
When dealing with polar equations, graphing becomes a fundamental step as it provides visual insight into the relationships defined by the equations. By mastering polar equations, students can gain a deeper understanding of complex mathematical concepts and their practical applications.

Coordinate Systems

Understanding different coordinate systems is crucial for solving problems in mathematics and related fields. The rectangular (Cartesian) coordinate system is based on a grid defined by perpendicular axes, while the polar coordinate system provides a circular framework defined by radius and angle.
Each system has its strengths. Rectangular coordinates excel with straightforward linear equations, while polar coordinates simplify problems involving circles and periodic motion. This flexibility allows mathematicians and scientists to choose the most effective system for a given problem.
By recognizing the characteristics and applications of each system, one can develop a versatile approach to problem-solving. This includes knowing when to leverage the radial symmetry of polar coordinates or the linear simplicity of rectangular coordinates, ultimately leading to clearer insights and solutions.