Question

Use the results of Exercises \(78-79\) to describe and graph the following circles. $$r^{2}-2 r(2 \cos \theta+3 \sin \theta)=3$$

Short answer

Question: Describe and graph the circle represented by the polar equation $$r^{2}-2 r(2 \cos \theta+3 \sin \theta)=3.$$ Answer: The circle has a center at (2,3) in Cartesian coordinates and a radius of √10. To graph the circle, plot the center point (2,3) on the coordinate plane and draw a circle with a radius of √10 around the center point.

Step-by-step solution

Identify the polar equation of the circle

The given polar equation is $$r^{2}-2 r(2 \cos \theta+3 \sin \theta)=3.$$ We will use this equation to find the center and radius of the circle.

Determine the center of the circle

According to Exercises 78-79, for a polar equation of the form $$r^2 - 2r(A\cos\theta + B\sin\theta) = C$$, the center of the circle (\((h, k)\) in Cartesian coordinates) is given by: \((h, k) = (A, B)\). Therefore, the center of the given circle is \((2,3)\).

Determine the radius of the circle

The radius of the circle for the given polar equation is given by the formula: $$\text{radius} = \sqrt{A^2 + B^2 - C}.$$ Substitute the values of \(A\), \(B\), and \(C\) to find the radius: $$\text{radius} = \sqrt{2^2 + 3^2 - 3} = \sqrt{4 + 9 - 3} = \sqrt{10}.$$ The radius of the circle is \(\sqrt{10}\).

Describe the circle

The circle has a center at \((2,3)\) in Cartesian coordinates and a radius of \(\sqrt{10}\).

Graph the circle

To graph the circle, first plot the center point \((2,3)\) on the coordinate plane. Then, draw a circle with a radius of \(\sqrt{10}\) around the center point. The resulting circle represents the graph of the polar equation $$r^{2}-2 r(2 \cos \theta+3 \sin \theta)=3.$$

Key concepts

Polar Equation

A polar equation is a way to express mathematical relationships or shapes using polar coordinates. In polar coordinates, each point is represented by a radius and an angle. This system is different from the Cartesian coordinate system, which uses x and y coordinates. In our exercise, the given polar equation is \(r^{2}-2r(2\cos\theta + 3\sin\theta) = 3\). Understanding polar equations allows us to describe various shapes, like circles, in a form that can be more convenient depending on the problem at hand.
The polar equation in this example helps us find the parameters we need to graph the circle, such as its center and radius, by relating it to specific constants \(A\), \(B\), and \(C\).
Key things to remember about polar equations include:

• The terms often involve trigonometric functions like cosine and sine, which relate to the angle \(\theta\).
• Polar equations can be converted into Cartesian forms for easier interpretation when needed.

Circle Graphing

Graphing circles from polar equations involves determining two main aspects: the circle's center and radius. In this particular problem, we're given a polar equation and asked to graph a circle.
To start graphing:

• We first determine the center by identifying the constants associated with \(\cos\theta\) and \(\sin\theta\), which represent Cartesian coordinates \((h, k)\).
• Next, we calculate the circle's radius using the formula derived from these constants and the constant \(C\) in the polar equation.
• Finally, once the center (\(h, k\)) and radius are known, we can plot these on a coordinate plane and draw the circle.

This approach allows us to translate polar equations into easily visualized circles on a graph, bridging the gap between abstract equations and geometric shapes.

Mathematical Transformation

Mathematical transformations are key in converting and interpreting equations in various forms. By applying transformations, we can extract meaningful geometric information from abstract algebraic equations.
For example, consider the equation \(r^2 - 2r(A\cos\theta + B\sin\theta) = C\). Here, transformations allow us to determine:

• The center of the circle by equating \(A\) and \(B\) to the x and y coordinates.
• The radius by applying the formula \(\sqrt{A^2 + B^2 - C}\), which gives us a tangible measure of the circle's size.

These transformations are powerful tools, transforming a complex equation into easily understandable components like centers and radii of shapes.

Coordinate Conversion

Coordinate conversion involves shifting between different systems of representing points in space. In this case, from polar to Cartesian coordinates.
To convert a polar coordinate \((r, \theta)\) to Cartesian coordinates \((x, y)\), utilize the formulas:

• \(x = r \cos\theta\)
• \(y = r \sin\theta\)

This process is crucial for understanding and graphing shapes given in polar equations. For the exercise, conversion enabled us to find the circle's center in Cartesian coordinates: \((2, 3)\).
Coordinate conversion simplifies complex analyses, making graphical interpretations straightforward and understandable, turning equations into visual, geometric forms.