Question

Rewrite the equation \(r=\sec \theta \tan \theta\) in rectangular coordinates and identify its graph.

Short answer

The equation in rectangular coordinates is \(x^4 = x^2 y^2 + y^4\), representing intersecting lines.

Step-by-step solution

Recall Coordinate Transformations

To convert between polar and rectangular coordinates, we use the transformations:\[ x = r \cos \theta \]\[ y = r \sin \theta \]\[ r = \sqrt{x^2 + y^2} \]\[ \tan \theta = \frac{y}{x} \]

Express in Terms of Sine and Cosine

Rewrite the given polar equation in terms of sine and cosine:\[ r = \sec \theta \tan \theta = \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{\cos \theta} = \frac{\sin \theta}{\cos^2 \theta} \]

Substitute Coordinate Transformations

Now substitute the expressions for \( \sin \theta \) and \( \cos \theta \) in terms of rectangular coordinates:\[ \sin \theta = \frac{y}{r} \quad \text{and} \quad \cos \theta = \frac{x}{r} \]Therefore,\[ r = \frac{\frac{y}{r}}{\left(\frac{x}{r}\right)^2} = \frac{y \cdot r^2}{x^2} \]

Simplify the Equation

Multiply both sides by \(x^2\) to clear the fraction:\[ r \cdot x^2 = y \cdot r^2 \]Divide both sides by \(r\) and simplify:\[ x^2 = y \cdot r \]Now substitute \( r = \sqrt{x^2 + y^2} \):\[ x^2 = y \cdot \sqrt{x^2 + y^2} \]

Square Both Sides

To remove the square root, square both sides:\[ x^4 = y^2 (x^2 + y^2) \]Expand and rearrange:\[ x^4 = x^2 y^2 + y^4 \]

Recognize the Graph Type

The resulting equation \( x^4 = x^2 y^2 + y^4 \) is a quartic equation that can be recognized as the equation of two intersecting straight lines or a degenerate curve. Solving further or visually analyzing this indicates it represents a pair of lines that meet at the origin.

Key concepts

Polar Coordinates

Polar coordinates represent points in a plane using distance and angle. They are denoted as \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. Unlike rectangular coordinates \((x, y)\), which use horizontal and vertical distances, polar coordinates focus on direction and magnitude.
Polar coordinates are especially useful in scenarios involving circular or rotational symmetry, such as physics problems involving circular motion.

• Angle \(\theta\): Measured in degrees or radians, indicating rotation from the positive x-axis.
• Radius \(r\): Distance from the origin; can be negative, indicating direction opposite to \(\theta\).

Understanding how to transform between polar and rectangular coordinates is essential, as it allows solving problems in the most convenient format.

Coordinate Transformations

Transforming between polar and rectangular coordinates involves simple equations. These transformations help convert expressions like \(r = \sec \theta \tan \theta\) into more familiar forms in rectangular coordinates. Let's break down the essential transformations:

• Rectangular Coordinates: \((x, y)\)
• Equations for transformation:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\tan \theta = \frac{y}{x}\)

These equations allow for seamless conversion between polar and rectangular systems, making it easier to analyze or plot complex equations. When transforming, it's crucial to understand the geometric meaning behind each expression to identify appropriate graphical representations.

Quartic Equation

A quartic equation is a polynomial equation of degree four. The standard form is \(ax^4 + bx^3 + cx^2 + dx + e = 0\). In the exercise solution, by transforming the polar equation into rectangular coordinates, we've derived the quartic equation:\[ x^4 = x^2 y^2 + y^4 \]These equations can represent various graphs, such as pairs of lines or more complex shapes.
Quartic equations are significant because:

• They can describe intersections of lines.
• They might represent ellipses, parabolas, or other curves, depending on their form.
• Understanding their structure helps in graph analysis and calculus problems.

In this exercise, the equation describes a degenerate form, resulting in intersecting lines at the origin. By recognizing the graph type, we can better interpret the mathematical relationships that the equation encodes.