Problem 46 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r^{2}=-6 r \sin \theta$$

Step-by-Step Solution

Answer

The graph is a circle centered at (0, -3) with radius 3.

Step 1: Convert the Polar Equation to Standard Form

To begin converting the given polar equation, \(r^{2} = -6r \sin \theta\), recognize that \(r^{2}\) can also be expressed using the polar coordinate transformation formulas in Cartesian form. The expression \(-6r \sin \theta\) can sometimes lead to recognizing known geometric forms in Cartesian coordinates.

Step 2: Apply Polar to Cartesian Coordinate Conversion Formulas

Use the conversions: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, remember that \(r = \sqrt{x^2 + y^2}\) and \(r \sin \theta = y\). By substituting \(y\) for \(r \sin \theta\) in this equation, \(r^2 = -6r \sin \theta\) becomes \(r^2 = -6y\).

Step 3: Substitute \(r = \sqrt{x^2 + y^2}\)

Replace \(r^2\) with \((x^2 + y^2)\) to convert the equation entirely to Cartesian coordinates. This results in \(x^2 + y^2 = -6y\).

Step 4: Organize into a Standard Circle Equation

Rearrange the equation to get it closer to the standard circle form by bringing all terms to one side: \(x^2 + y^2 + 6y = 0\).

Step 5: Complete the Square to Identify the Graph

Complete the square for the \(y\)-terms: add \((\frac{6}{2})^2 = 9\) to both sides. This transforms the equation to \(x^2 + (y + 3)^2 = 9\).

Step 6: Identify the Graph as a Circle

The equation \(x^2 + (y + 3)^2 = 9\) is now in the form \((x - h)^2 + (y - k)^2 = r^2\), which represents a circle in Cartesian coordinates with center at \((h, k)\) and radius \(r\). Thus, the graph is a circle centered at \((0, -3)\) with a radius of 3.

Key Concepts

Polar Coordinates

Polar coordinates are a unique way to represent the position of a point on a plane. Instead of using the standard x and y, or horizontal and vertical placements as in Cartesian coordinates, polar coordinates use a distance and an angle.

Each point in a polar system is given by:

• \(r\): the radius, or distance from the origin to the point.
• \(\theta\): the angle formed with the positive x-axis, measured in radians.

For example, a point at \((r, \theta) = (5, \frac{\pi}{4})\) translates to a point that is 5 units away from the origin and formed at an angle of \(45^\circ\).
Using polar coordinates can simplify complex equations, especially when they deal with rotation or circular motion.

Cartesian Coordinates

Cartesian coordinates represent a point in the plane using two perpendicular axes, usually labeled x and y. These coordinates are the most common way of plotting points and drawing shapes.

In a Cartesian system:

• The x-coordinate measures horizontal displacement from the origin.
• The y-coordinate measures vertical displacement from the origin.

A Cartesian coordinate system is beneficial because it provides a simple way to visualize mathematical concepts, especially those related to straight lines or linear transformations.

Equation Transformation

Equation transformation is the process of converting an equation from one form to another, preserving its truth but changing its appearance or format. In the context of this exercise, we deal with transforming polar equations to Cartesian equations.

The key steps in these transformations include:

• Using the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\).
• Substituting \(r = \sqrt{x^2 + y^2}\) to replace polar terms with Cartesian terms.

These transformations allow us to see geometric shapes more clearly in a way we’re familiar with—often revealing circles, ellipses, or other conic sections.

Circle Equation

The circle equation in Cartesian coordinates takes the standard form:\[(x - h)^2 + (y - k)^2 = r^2\]where \((h, k)\) is the center of the circle and \(r\) is the radius.

To convert an equation to this form, it's essential to rearrange and simplify terms. In our example, by converting and reorganizing the equation, we gained \(x^2 + (y + 3)^2 = 9\). This tells us there’s a circle centered at \((0, -3)\) with a radius of 3.

Understanding this form is crucial for graphing and recognizing the shape and position of a circle in a plane.

Completing the Square

Completing the square is a method used to convert a quadratic expression into a perfect square format. This technique is particularly useful in rewriting circles as standard forms in Cartesian coordinates.

For a term like \(y^2 + 6y\), you complete the square by:

• Calculating \((\frac{6}{2})^2 = 9\).
• Adding 9 to both sides of the equation to form \( (y + 3)^2 \).

By doing so, the expression is now simplified to show how it factors neatly into a square, which is a pivotal step in recognizing the geometric form like a circle or ellipse. This technique is widely used across several areas of algebra to aid in solving quadratic equations and graphing parabolas effectively.