Problem 60 - College Algebra

Question

Recall from Rotation of Axes that equations of conics with an \(x y\) term have rotated graphs. For the following exercises, express each equation in polar form with \(r\) as a function of \(\theta\). $$ 2 x y+y=1 $$

Step-by-Step Solution

Answer

r = \frac{1}{\sin \theta (2 \cos \theta + 1)}

Step 1: Identify the transformation formulas

To convert from Cartesian coordinates to polar coordinates, we use:\[ x = r \cos \theta \] \[ y = r \sin \theta \]

Step 2: Substitute the polar expressions

Replace \(x\) and \(y\) in the equation \(2xy + y = 1\) with \(x = r \cos \theta\) and \(y = r \sin \theta\):\[2(r \cos \theta)(r \sin \theta) + r \sin \theta = 1\]

Step 3: Simplify the equation

To simplify the equation, factor out \(r\sin\theta\):\[r \sin \theta (2r \cos \theta + 1) = 1\]

Step 4: Solve for \(r\)

Isolate \(r\) on one side to express \(r\) as a function of \(\theta\):\[ r = \frac{1}{\sin \theta (2 \cos \theta + 1)} \]

Key Concepts

Rotation of Axes

Understanding the rotation of axes is crucial when dealing with equations of conics that have cross-product terms like \(xy\). Rotation of axes is a coordinate transformation technique used to simplify these equations.

This technique involves rotating the entire coordinate system by a certain angle to eliminate the \(xy\) term. In simpler terms, you're changing the viewpoint to simplify the problem. This makes it easier to identify and work with conic sections such as ellipses, parabolas, or hyperbolas.

In polar coordinates, this is achieved by expressing the \(x\) and \(y\) terms in terms of \(r\) and \(\theta\). This transformation allows for a more straightforward visualization. Rotation of axes is thus a useful tool to make complex conic equations more tractable.

Conics

Conic sections are curves formed by intersecting a plane with a double-napped cone. The main types of conic sections include circles, ellipses, parabolas, and hyperbolas.

Each type of conic has a distinct equation in Cartesian coordinates. However, when the equation includes an \(xy\) term, it often indicates that the conic's axes are not aligned with the usual coordinate axes.

• Circles: Equations of the form \(x^2 + y^2 = r^2\).
• Ellipses: Look like distorted circles.
• Parabolas: Have equations like \(y = ax^2 + bx + c\).
• Hyperbolas: Consist of two disconnected curves.

By using the rotation of axes or converting to polar coordinates, you can better understand the nature and orientation of these conic sections. This approach simplifies the identification and manipulation of these curves.

Cartesian Coordinates

Cartesian coordinates are the most common system used in geometry, involving two perpendicular axes, typically labeled as \(x\) and \(y\). Points are defined based on their distance from these axes.

In Cartesian coordinates, equations represent geometrical figures. For example, lines, circles, and parabolas can all be drawn using these coordinates. However, when you encounter equations with cross terms, their representation might cause confusion under the Cartesian system.

• The equation \(2xy + y = 1\) is hard to interpret directly in the Cartesian system.
• Transforming to a different system like polar coordinates or rotating the axes can provide clarity.

Interpreting such equations may require transforming them to a more intuitive system. Cartesian coordinates serve as a baseline but often necessitate adjustments for complex shapes and equations.

Coordinate Transformation

Coordinate transformations involve changing the reference framework to make a problem easier to solve or understand. The classic transformation from Cartesian to polar coordinates is a prime example.

Transformations are particularly helpful when dealing with equations that do not neatly fit into one coordinate system. By switching coordinates, complex relationships can often be simplified.

• To convert from Cartesian to polar, use: \(x = r \cos \theta\) and \(y = r \sin \theta\).
• This transformation makes it easier to express curves like conics in a simpler form.

Transformations can simplify math by providing new perspectives, aiding in visualizing and solving problems. Whether rotating axes in Cartesian systems or shifting to polar formats, these techniques are each powerful tools in mathematical exploration.