Problem 42 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r^{2}=-6 r \sin \theta $$

Step-by-Step Solution

Answer

The equation represents a circle with center \((0, -3)\) and radius 3.

Step 1: Recognize the Polar Equation

The given polar equation is \( r^2 = -6r \sin \theta \). We need to convert this into a Cartesian equation form \((x, y)\).

Step 2: Substitute Polar to Cartesian Coordinates

In polar coordinates, \( r^2 = x^2 + y^2 \), \( x = r \cos \theta \), and \( y = r \sin \theta \). We can substitute \( y = r \sin \theta \) into the equation: \( r^2 = -6y \).

Step 3: Use the Identity for \(r^2\)

Replace \( r^2 \) with \( x^2 + y^2 \) in the equation: \( x^2 + y^2 = -6y \).

Step 4: Rearrange to Standard Form

Rearrange the equation to get a more recognizable form: \( x^2 + y^2 + 6y = 0 \).

Step 5: Complete the Square

To identify the graph, complete the square for the \( y \) term. We can rewrite \( y^2 + 6y \) as \((y + 3)^2 - 9\). Thus, the equation becomes \( x^2 + (y + 3)^2 = 9 \).

Step 6: Identify the Graph

The equation \( x^2 + (y + 3)^2 = 9 \) represents a circle with center at \((0, -3)\) and radius 3.

Key Concepts

Cartesian coordinates

In geometry, Cartesian coordinates allow us to pin down any point on a plane with precision. They use a pair of values \(x, y\) to indicate a location concerning the horizontal x-axis and the vertical y-axis. These coordinates form the backbone of graphing and help visualize geometric shapes by plotting points, lines, and curves.
Here's why Cartesian coordinates are vital:

• They provide a clear, unambiguous way to determine the position of a point.
• Using these coordinates, we can express curves and geometric figures algebraically.
• Transformations, such as rotations or translations, are easily handled.
• Equations expressed in this coordinate system are straightforward to differentiate and integrate.

Understanding how to convert polar coordinates to Cartesian ones, as in the original problem, allows for enriched analysis of geometric entities like circles, ellipses, or even more complex graphs.

Polar coordinates

Polar coordinates present a different approach to identifying a point on a plane compared to Cartesian coordinates. Instead of measuring horizontal and vertical distances, polar coordinates use \(r\), the radial distance from the origin, and \(\theta\), the angular orientation from the positive x-axis.
This system is especially useful in scenarios involving circular or rotational motion, or phenomena that naturally have a center point:

• They simplify equations describing circular paths and spirals.
• They are beneficial in calculating angles and lengths where circles are involved.
• In physics, they can simplify the analysis of systems with circular symmetry.

To transition between polar and Cartesian systems, use the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \), as shown in the exercise. This transition offers flexibility in choosing the more suitable framework for a given mathematical problem or real-world application.

Completing the square

Completing the square is a valuable algebraic method used to rewrite a quadratic equation into a perfect square form. This technique is instrumental in facilitating the identification of certain characteristics of the equation, such as the vertex of a parabola or center of a circle.
Here's a simple breakdown of how completing the square works:

• Identify the quadratic and linear terms, such as \(y^2 + 6y\).
• Take half of the linear coefficient (in this case, 6), then square it \((3^2 = 9)\).
• Add and subtract this square inside the equation to preserve equality, resulting in \((y + 3)^2 - 9\).

This reformulation is crucial in transforming the given equation into one that represents a circle clearly, helping identify the center \(0, -3\) and radius 3 with ease. This technique is also applied in conics and solving quadratic equations.

Circle equations

A circle’s equation in Cartesian coordinates is typically expressed in the form \( (x - a)^2 + (y - b)^2 = r^2 \), representing a circle centered at \(a, b\) with radius \(r\). Recognizing this format helps quickly understand and sketch circles based on their equations.
Key points about circle equations:

• The terms \(a\) and \(b\) indicate the horizontal and vertical offsets from the origin, respectively.
• The radius \(r\) determines how far the boundary of the circle is from the center.
• Completing the square is often used to rearrange any general quadratic equation into the standard circle equation.
• Once transformed, the circle's properties are immediately visible, enabling straightforward analysis or graphing.

In the provided exercise, after converting the polar equation to Cartesian form and completing the square, we discerned the equation \(x^2 + (y + 3)^2 = 9\), pinpointing a circle with center at \(0, -3\) and radius 3.