Problem 19 - College Algebra

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r=\frac{2}{5-3 \sin \theta} $$

Step-by-Step Solution

Answer

The rectangular equation is \(x^2 + y^2 - 4x + 5 = 0\).

Step 1: Identify Conic Section Components

The given polar equation is \( r = \frac{2}{5 - 3\sin \theta} \). We identify that it represents a conic section with \( e = 3 \) and \( d = 2 \). The conic section type can be determined by the eccentricity: since \( e = 3 > 1 \), it's a hyperbola.

Step 2: Substitute Polar to Rectangular Relations

Recall the relationships \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \). Substitute these into the polar equation: \[ r = \frac{2}{5 - 3\frac{y}{r}}. \] Simplifying, we get: \( r(5 - 3\frac{y}{r}) = 2 \).

Step 3: Clear the Fraction

To eliminate the fraction, multiply through by \( r \): \( r^2 = 2r - 3y \). This can be written as \( r^2 = 2r - 3y \).

Step 4: Replace \(r\) with Rectangular Form

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

Step 5: Solve for a Rectangular Equation

Square both sides to clear the square root: \((x^2 + y^2)^2 = 4(x^2 + y^2) - 12y\sqrt{x^2 + y^2} + 9y^2\). Expand and rearrange terms to reach: \(x^2 + y^2 - 4x + 5 = 0\).

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to the traditional rectangular (or Cartesian) coordinate system. Instead of using coordinates in terms of x and y, they use a radius and angle to locate points on a plane.

• Radius ( ): This is the distance from a reference point, known as the pole or origin, to any point in the plane.
• Angle ( heta): Measured in radians, this is the angle from a reference direction, typically the positive x-axis, to the line connecting the point to the pole.

This system is particularly useful in situations where relationships are more naturally expressed using angles and distances, such as in circular and spiral structures. For example, in the polar equation given in the problem, the position of each point is determined by the equation involving and heta.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most commonly used coordinate system in mathematics. This system describes a point in space by its distances along perpendicular axes, usually labeled as x (horizontal axis) and y (vertical axis). Some of the key features include:

• x-coordinate: Indicates the horizontal position from the origin.
• y-coordinate: Indicates the vertical position from the origin.

Rectangular coordinates are particularly effective for describing linear and quadrilateral shapes, making them versatile for various types of equations. Converting from polar coordinates often involves trigonometric relations like \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \). These identities allow for the translation of curves described in polar form into a rectangular equation as shown in the solution process.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The main types include circles, ellipses, parabolas, and hyperbolas. Their definition hinges on a few key properties:

• Circle: A special case of an ellipse with eccentricity (\( e \)) equal to 0.
• Ellipse: Has an eccentricity between 0 and 1, representing a closed curve.
• Parabola: Represents a curve where the eccentricity is exactly 1.
• Hyperbola: Characterized by an eccentricity greater than 1, representing an open curve.

These figures can be described in both polar and rectangular formats, providing a rich field of study in geometry and algebra. In polar coordinates, the conic sections are defined with equations linking radius and angles, whereas in rectangular coordinates, the equations involve x and y variables, often leading to quadratic forms.

Hyperbola

A hyperbola is one type of conic section formed by the intersection of a plane with both naps of a cone. It consists of two disconnected curves called branches that mirror each other. Key characteristics of a hyperbola include:

• Asymptotes: Lines that the curve approaches but never touches.
• Vertices: Points where the hyperbola intersects its principal axis.
• Foci: Two fixed points situated inside each branch of the curve.

Hyperbolas are defined by an eccentricity greater than 1. This conic is particularly interesting because of its real-world applications, such as understanding the paths of objects under gravitational forces in astronomy or the representation of reciprocal relationships in physics. Identifying a hyperbola from a polar equation involves recognizing that the eccentricity (\( e \)) is greater than 1, as solved in the exercise where \( e = 3 \).

Eccentricity

Eccentricity is a fundamental concept in the study of conic sections, describing the shape of the conic. It is a non-negative real number that uniquely identifies the type of conic and some of its properties. The values of eccentricity define conics as follows:

• Circle: \( e = 0 \), a perfectly round figure.
• Ellipse: \( 0 < e < 1 \), creating an oval shape.
• Parabola: \( e = 1 \), which forms a symmetrical, open curve.
• Hyperbola: \( e > 1 \), results in an open curve with two branches.

Eccentricity is crucial when transforming a polar equation of a conic into its rectangular form. This transformation retains the properties of the conic in terms of its eccentricity, as shown in the equation \( r = \frac{2}{5-3 \sin \theta} \). Here, by identifying \( e=3 \), it is clear that the conic is a hyperbola.