Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, convert the polar equation of a conic section to a rectangular equation. \(r=\frac{6 \csc \theta}{3+2 \csc \theta}\)

Step-by-Step Solution

Answer

The rectangular equation is \(3y\sqrt{x^2 + y^2} = 6 - 2(x^2 + y^2)\).

Step 1: Understand the Polar Equation

The given polar equation is \( r = \frac{6 \csc \theta}{3 + 2 \csc \theta} \). In this equation, \( \csc \theta \) is the cosecant function, which is the reciprocal of sine; thus, \( \csc \theta = \frac{1}{\sin \theta} \). Let's rewrite the function in terms of sine.

Step 2: Substitute Cosecant

Substitute \( \csc \theta \) with \( \frac{1}{\sin \theta} \). The equation becomes: \[r = \frac{6 \cdot \frac{1}{\sin \theta}}{3 + 2 \cdot \frac{1}{\sin \theta}} = \frac{6}{\sin \theta (3 + \frac{2}{\sin \theta})} = \frac{6}{3\sin \theta + 2}.\]

Step 3: Express in Terms of \(x\) and \(y\)

Use the polar-to-rectangular coordinate conversions: \( r = \sqrt{x^2 + y^2} \) and \( \sin \theta = \frac{y}{r} \). Substitute these into the equation: \[ r = \frac{6}{3 \frac{y}{r} + 2}, \] which simplifies to \[ r = \frac{6}{\frac{3y}{r} + 2}. \] Multiply through by \(r\) to remove the fraction: \(r^2(3y + 2r) = 6r\).

Step 4: Simplify the Expression

Further simplify the equation: \[3r^2y + 2r^3 = 6r.\] Divide through by \(r\): \[3ry + 2r^2 = 6.\] Now replace \(r\) with \(\sqrt{x^2 + y^2}\): \[3y\sqrt{x^2 + y^2} + 2(x^2 + y^2) = 6.\]

Step 5: Find the Rectangular Equation

Rearrange the terms to get the rectangular equation: \[3y\sqrt{x^2 + y^2} = 6 - 2(x^2 + y^2).\] This is the rectangular form of the conic section derived from the original polar equation.

Key Concepts

Polar Coordinates

Polar coordinates are a way of expressing points in a plane through angles and distances from a central point called the pole (similar to the origin in Cartesian coordinates). In polar coordinates, each point is determined by:

• Radius (\( r \)), which is the distance from the pole.
• Angle (\( \theta \)), which measures the angle from the positive x-axis (considered as 0 degrees).

The polar coordinate system is especially useful in situations where a relationship is naturally expressed in terms of angles and distances, such as in circular and rotational systems. In our exercise, the equation is given in polar form as \( r = \frac{6 \csc \theta}{3 + 2 \csc \theta} \).
This requires understanding trigonometric functions like cosecant, denoted as \( \csc \theta \), which is the reciprocal of sine (\( \csc \theta = \frac{1}{\sin \theta} \)). Our task involves converting this equation into rectangular coordinates for better understanding in a typical Cartesian framework.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in the plane using two values:

• \( x \): the horizontal distance from the origin.
• \( y \): the vertical distance from the origin.

Using these values often provides a straightforward way to understand geometric figures and their properties. Changing from polar to rectangular coordinates involves conversions:

• Using \( r = \sqrt{x^2 + y^2} \) to relate the distances in the plane.
• Replacing trigonometric functions based on their relationships with \( x \) and \( y \), like \( \, \sin \theta = \frac{y}{r} \, \).

This flexibility in switching between coordinate systems is especially helpful for analyzing equations of shapes such as conic sections. In our task, the aim is to translate the expression into this format so that it can reveal its shape within the Cartesian plane.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane, which gives rise to different shapes, such as circles, ellipses, parabolas, and hyperbolas. These curves are described well in both polar and rectangular coordinates but are often more familiar in rectangular form.
In the exercise, the original polar equation represents one such conic section, which when converted to rectangular coordinates resulted in:\[3y\sqrt{x^2 + y^2} = 6 - 2(x^2 + y^2).\]This expression fits the rectangular format where the relationship between \( x \) and \( y \) is clearly established, typical of different types of conics.

• Converting from polar to rectangular in this context helps in plotting and visualizing these conic sections in a standard plane, providing insights into their structure and behavior.
• Understanding conic sections and their equations is crucial in fields such as physics, engineering, and astronomy, where they model various natural phenomena.

By learning these conversions, students gain a deeper comprehension of geometric shapes and their properties.