Problem 22 - College Algebra

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{4}{2+2 \sin \theta} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the polar equation

The given polar equation, \(r = \frac{4}{2 + 2\sin\theta}\), describes a conic section with the positive direction of the y-axis as its directrix due to the \(\sin\theta\) term.

Step 2: Simplify the equation

Start by simplifying the polar equation: \(r = \frac{4}{2 + 2\sin\theta} = \frac{2}{1 + \sin\theta}\). Observe that it has the standard form for a conic in polar coordinates: \(r = \frac{ed}{1 + e\sin\theta}\). Here, \(e = 1\) and \(d = 2\).

Step 3: Convert to rectangular coordinates

Use the conversion relationships: \(x = r\cos\theta\), \(y = r\sin\theta\), and \(r^2 = x^2 + y^2\). According to the identity, \(\sin\theta = \frac{y}{r}\), substitute into the equation: \(r(1 + \frac{y}{r}) = 2\), which becomes \(r + y = 2\).

Step 4: Solve for r

From \(r + y = 2\), solve for \(r\): \(r = 2 - y\). Use the relation \(r^2 = x^2 + y^2\) to substitute \(r\): \((2 - y)^2 = x^2 + y^2\).

Step 5: Expand and rearrange

Expand the equation: \(4 - 4y + y^2 = x^2 + y^2\). Rearrange terms to get a standard form: \(x^2 + y^2 = 4y - 4\).

Step 6: Finalize the rectangular equation

Subtract \(y^2\) from both sides: \(x^2 = 4y - 4 - y^2\) which simplifies to \(x^2 + y^2 - 4y = -4\).

Key Concepts

Conic Sections

Conic sections are fascinating curves that are formed by the intersection of a plane and a cone. The primary types of conic sections include circles, ellipses, parabolas, and hyperbolas. Each has unique characteristics and equations that define its shape. Conic sections are crucial in mathematics and physics due to their prevalence in natural phenomena and their application in diverse areas such as orbital motion and architecture.

When dealing with conic sections, having a conic in polar form, like in this exercise, indicates how the curve is oriented relative to a central point. Recognizing these curves in polar coordinates often helps us understand their geometric properties and relationships better. When we convert these equations into rectangular form, we can describe their shapes in the familiar Cartesian plane.

Polar Coordinates

Polar coordinates provide a unique way of plotting points using a radius and an angle. Originating from a central point, every point in polar coordinates is defined based on:

• The distance from the origin, known as the radius, denoted by \( r \).
• The angle, \( \theta \), formed with the positive direction of the x-axis.

The polar coordinate system is especially useful in fields like engineering and physics due to its ability to simplify the mathematics of curves and circular motions.

Understanding the polar equation within this context is essential when converting to rectangular coordinates. Equations written in polar form often reveal symmetries and important features of the curve, such as directrix and focus, which may not be immediately evident in Cartesian form. In the given equation, \( r = \frac{4}{2 + 2\sin\theta} \), the angle \( \theta \) affects the shape of the conic, thereby characterizing it.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, use the familiar x- and y-axes to locate points in two-dimensional space. They are characterized by:

• The x-coordinate, representing horizontal position.
• The y-coordinate, representing vertical position.

This coordinate system provides a straightforward way to plot points and visualize functions on a graph.

In mathematics, being able to convert between coordinate systems such as polar to rectangular is vital. It allows us to express our results in the most suitable form for further analysis or application. For this exercise, converting an equation from polar to rectangular format helps us to place the conic within familiar x-y coordinates, making it easier to visualize and interpret in graphical form.

Conversion Relationships

Conversion between polar and rectangular coordinates involves a few key relationships which bridge the two systems:

• \( x = r\cos\theta \) and \( y = r\sin\theta \), which directly translate polar coordinates \((r, \theta)\) into rectangular coordinates \((x, y)\).
• The relationship \( r^2 = x^2 + y^2 \) often helps in translating expressions involving \( r \) in polar equations into a rectangular format.
• Trigonometric identities, like \( \sin\theta = \frac{y}{r} \) and \( \cos\theta = \frac{x}{r} \), facilitate these transformations, especially in equations involving trigonometric functions.

Using these relationships, the polar equation \( r = \frac{2}{1 + \sin\theta} \) is converted into a rectangular form, enabling us to derive the equation \( x^2 + y^2 - 4y = -4 \). This final form represents the conic section in Cartesian coordinates, ready for analysis and visualization.