Problem 18 - College Algebra

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{4}{1+3 \sin \theta} $$

Step-by-Step Solution

Answer

The rectangular equation is \(x^2 - 8y^2 + 24y = 16\).

Step 1: Understand the Polar Equation of the Conic Section

The given equation is \(r = \frac{4}{1 + 3\sin\theta}\). It is a polar equation of a conic section with the form \(r = \frac{ed}{1 + e\sin\theta}\), where \(e\) is the eccentricity. Here, \(e = 3\) and \(d = 4\). Since \(e > 1\), the conic is a hyperbola.

Step 2: Recall the Polar to Rectangular Conversion Formulas

To convert from polar to rectangular coordinates, we use the relationships \(x = r\cos\theta\), \(y = r\sin\theta\), \(r^2 = x^2 + y^2\), and \(\tan\theta = \frac{y}{x}\). These will be used to express the equation in rectangular form.

Step 3: Substitute the Sine Component

The equation has the term \(\sin\theta\), which can be rewritten in terms of \(y\). We have \(\sin\theta = \frac{y}{r}\), thus substitute this into the equation: \[ r = \frac{4}{1 + 3\left(\frac{y}{r}\right)} \] which simplifies to \[ r = \frac{4}{1 + \frac{3y}{r}} \].

Step 4: Clear the Denominator to Eliminate \(r\)

Multiply through by \(r\) to eliminate the fraction: \[ r(1 + \frac{3y}{r}) = 4 \] which simplifies to \[ r + 3y = 4 \]. Since \(r = \sqrt{x^2 + y^2}\), substitute it back: \[ \sqrt{x^2 + y^2} + 3y = 4 \].

Step 5: Rearrange and Simplify

Solve for \(\sqrt{x^2 + y^2}\) in terms of \(y\): \[ \sqrt{x^2 + y^2} = 4 - 3y \]. Square both sides to eliminate the square root: \[ x^2 + y^2 = (4 - 3y)^2 \] which expands to \[ x^2 + y^2 = 16 - 24y + 9y^2 \].

Step 6: Rearrange into Standard Rectangular Form

The equation from Step 5 is \[ x^2 + y^2 = 16 - 24y + 9y^2 \]. Rearrange to get it into a more standard form: \[ x^2 + y^2 - 9y^2 + 24y = 16 \]. Simplifying further gives \[ x^2 - 8y^2 + 24y = 16 \].

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double cone at different angles. These can be circles, ellipses, parabolas, or hyperbolas. Each type of conic section has unique properties and equations.

• Circle: A special ellipse with equal axes.
• Ellipse: A closed curve, forming an elongated circle.
• Parabola: A curve with a single line of symmetry, formed when the plane is parallel to the cone's side.
• Hyperbola: An open curve with two branches, formed when the plane intersects both halves of the cone.

Conic sections have diverse applications, from orbital paths of planets to the design of optical lenses. The exercise focuses on converting a polar equation of one such section, a hyperbola, to a rectangular form.

Polar Coordinates

Polar coordinates use a point's distance from a reference point and its angle from a reference direction to define its position. Unlike rectangular coordinates, which use horizontal (x) and vertical (y) distances:

• Each point in the plane is determined by a radius, \(r\), and an angle, \(\theta\).
• \(r\) is the distance from the origin, and \(\theta\) is the angle measured counterclockwise from the positive x-axis.

Polar coordinates are especially useful in contexts where symmetry and angles play a significant role, such as in circular or angular patterns. They often simplify the mathematics of curves like conic sections.

Rectangular Coordinates

Rectangular (or Cartesian) coordinates describe points using horizontal and vertical distances from a fixed reference point, the origin. This system uses:

• A pair of coordinates, \((x, y)\), to define every point's exact position.
• \(x\) measures the horizontal distance, while \(y\) measures the vertical distance from the origin.

In mathematical transformations, like converting from polar to rectangular coordinates, relationships such as \(x = r\cos\theta\) and \(y = r\sin\theta\) help in expressing equations in a linear and familiar form, often making further calculus operations simpler to handle.

Hyperbola

A hyperbola consists of two mirrored open curves that replicate the figure of a parabola but diverge. It is one of the classic conic sections, characterized by:

• An eccentricity \(e > 1\).
• An equation of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) in rectangular coordinates.
• Certain symmetry properties, usually around two axes.

The hyperbola represented in polar form often reveals how the curve opens relative to the origin and what kind of transformation is needed to express it in rectangular coordinates. Understanding these aspects allows students to better visualize and solve problems involving complex curves in geometry and physics.