Problem 21 - College Algebra with Corequisite Support

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{3}{2+5 \quad \cos \theta}\)

Step-by-Step Solution

Answer

Convert the polar equation using coordinate relationships to obtain a rectangular form.

Step 1: Identify the Polar Equation

The given polar equation is \(r = \frac{3}{2 + 5 \cos \theta}\). This is a conic section with eccentricity \(e\) and directrix \(d\). In the standard polar form \(r = \frac{ed}{1 + e \cos \theta}\), the equation matches. By comparing, we find that \(ed = 3\), \(e = 5\), and \(d = \frac{3}{5}\).

Step 2: Use Relationships between Polar and Rectangular Coordinates

Recall the relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), and \(r^2 = x^2 + y^2\). Also, \(\cos \theta = \frac{x}{r}\) can be substituted where needed. We use these to convert the polar equation.

Step 3: Substitute for \(\cos \theta\)

Substitute \(\cos \theta = \frac{x}{r}\) into the polar equation: \[ r = \frac{3}{2 + 5 \left(\frac{x}{r}\right)} \]. This equation needs to be simplified further to eliminate \(r\) on the right-hand side.

Step 4: Clear the Fraction by Multiplying by \(r\)

Multiply both sides of the equation by \(r\) to clear the fraction: \[ r^2 = 3r - 5xr \]. Now we have \(r^2\) on the left, allowing us to utilize the relationship \(r^2 = x^2 + y^2\).

Step 5: Substitute \(r^2 = x^2 + y^2\)

Substitute \(r^2\) in the equation with \(x^2 + y^2\): \[ x^2 + y^2 = 3r - 5xr \]. To simplify, we express \(r\) fully in terms of \(x\) and \(y\).

Step 6: Substitute \(r = \sqrt{x^2 + y^2}\)

Substitute \(r = \sqrt{x^2 + y^2}\) into the equation:\[ x^2 + y^2 = 3\sqrt{x^2 + y^2} - 5x\sqrt{x^2 + y^2} \].

Step 7: Simplify the Equation

Finally, simplify and rearrange terms to obtain the rectangular form: \[ x^2 + y^2 = \left(3 - 5x\right)\sqrt{x^2 + y^2} \].To remove the square root, square both sides, or further simplification might be necessary depending on context, ending up with a convoluted algebraic expression that represents the conic in rectangular coordinates.

Key Concepts

Polar Equations

Polar equations are mathematical expressions where coordinates are given in terms of a radius and an angle. Unlike the rectangular system, which uses \(x\) and \(y\) coordinates, polar coordinates consist of a radius \(r\), representing the distance from the origin, and an angle \(\theta\), indicating direction.
A key aspect of polar equations is their efficiency in representing curves and shapes that originate or pivot around a central point.

• Example: The polar equation \(r = \frac{3}{2 + 5 \cos \theta}\) illustrates a relationship between \(r\) (radius) and \(\theta\) (angle).
• Conic Sections: Often, polar equations describe conic sections such as ellipses and hyperbolas, indicating orbits or paths conforming to these shapes.

Working with polar equations requires an understanding of trigonometric identities, as they are often used to convert between coordinate systems.

Rectangular Equations

Rectangular equations, also known as Cartesian equations, utilize the familiar coordinate system with \(x\) and \(y\) coordinates. This system is based on a horizontal and vertical axis intersecting at the origin. Converting a polar equation into a rectangular form can simplify the process of graphing the equation or solving related algebraic problems.
To convert a polar equation to a rectangular equation, we employ relationships like:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r^2 = x^2 + y^2\)

When you have expressions involving \(\cos \theta\), substitute \(\cos \theta = \frac{x}{r}\).
In the exercise, these substitutions help transform the polar equation into the rectangular form: \(x^2 + y^2 = (3 - 5x)\sqrt{x^2 + y^2}\). This process often involves clearing fractions and rationalizing expressions.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. Each resulting shape (ellipse, parabola, hyperbola) represents a unique set of geometric properties. These sections can be elegantly expressed both in polar and rectangular coordinates, each offering insights.

• Polar Representation: Simplifies the depiction of conics with a central point, using an equation like \(r = \frac{ed}{1 + e \cos \theta}\), where \(e\) is the eccentricity.
• Rectangular Conversion: Involves converting to a \(x\)- and \(y\)-coordinates format, making algebraic manipulation straightforward.

Understanding conic sections is crucial in many fields, from physics (planetary orbits) to engineering (antenna design), where these shapes frequently occur. Identifying the type of conic is determined by the eccentricity \(e\) and the geometric ratios in the equation.

Coordinate Systems

Coordinate systems are frameworks used to uniquely determine the position of points in a plane or space. They provide a reference for mathematical discussions and problem-solving.

• Polar Coordinates: Consist of a radius and angle, ideal for circular or rotational problems.
• Rectangular Coordinates: Use perpendicular lines (axes) for intuitive description of everyday objects.

In the context of converting polar equations like \(r= \frac{3}{2+5 \cos \theta}\) to a rectangular form, the idea is to translate a system based on angles and radii into one based on linear distances.
This conversion is not just an academic exercise but a practical methodology that finds usage in complex analyses, whether tackling curve drawings or dealing with three-dimensional modeling.