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Chapter 10: Problem 23

In Exercises 15-28, identify the conic and sketch its graph. \(r=\dfrac{3}{2+4\sin\ \theta}\)

Short Answer

Expert verified
The conic section represented by the polar equation \(r = 3 / (2 + 4\sin\ \theta)\) is a circle with its center at the origin, and radius 3.

Step by step solution

01

Converting Polar Equation into Cartesian Form

Begin by converting the polar equation into a Cartesian equation. Since the relationship between polar and Cartesian coordinates is given by \(x = r\cos\ \theta\) and \(y = r\sin\ \theta\), multiply \(r\) throughout the equation to have it in Cartesian form. The equation becomes \(r = x\cos\ \theta + y\sin\ \theta = 3\). Hence, the conic section is a circle.
02

Identify Key Features of the Circle

Next, identify the center and the radius of the circle. From the Cartesian equation form \(x^2 + y^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Apply this definition to the given equation \(x^2 + y^2 = 3^2\), it is clear that the circle has its center at the origin (h, k) = (0, 0) and its radius \(r = 3\).
03

Graph the Conic Section

Finally, sketch the graph of the circle. Draw a circle with its center at the origin and with radius 3. The graph represents the conic section of the given polar equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a way to represent points in a plane using a distance and an angle. In this system, any point is described by
  • the distance from a fixed point called the pole (similar to the origin in cartesian coordinates), often denoted as \(r\), and
  • an angle \(\theta\) that measures the direction from a reference direction, typically the positive x-axis.

Polar coordinates are especially useful in scenarios where the relationship between points is more about angles and distances rather than left-right and up-down. For instance, many natural phenomena are better described with them, like the motion of celestial objects.
In our example, we start with the equation \(r = \frac{3}{2+4\sin\ \theta}\), which is in polar form. To graph this in a more familiar x-y coordinate system, we convert it to cartesian coordinates.
Cartesian Coordinates
Unlike polar coordinates, cartesian coordinates use a grid-like system to locate points in 2D space, defined by:
  • an x-coordinate, representing the horizontal position, and
  • a y-coordinate, representing the vertical position.

The position of any point is given as an ordered pair \((x, y)\). This method is very intuitive because it corresponds to how most of us naturally describe movement and position - left/right and up/down.
To transition from polar to cartesian coordinates requires a little math. We use the formulas:
  • \(x = r\cos\ \theta\)
  • \(y = r\sin\ \theta\)

In our exercise, these conversions help us transform the polar equation into something easier to visualize and understand as a circle in the cartesian plane. We substitute these into the polar equation to get the cartesian equation \(x\cos\ \theta + y\sin\ \theta = 3\).
Circle Equation
A circle is one of the simplest curved shapes in geometry. In cartesian coordinates, the standard form of a circle's equation is:\[x^2 + y^2 = r^2\]where
  • \((h, k)\) are the coordinates of the circle's center, and
  • \(r\) is the radius of the circle.

When the center is at the origin \((0,0)\), the equation simplifies because \(h\) and \(k\) are zero. Hence, the equation becomes \(x^2 + y^2 = r^2\).
For the given problem, after converting the polar equation to cartesian form, it becomes clear that our conic section is a circle with center at \((0, 0)\) and radius 3 (since it matches the format \(x^2 + y^2 = 3^2\)). Recognizing the circle's equation helps in sketching its graph by marking the center at the origin and drawing a concentric loop with the given radius.

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Most popular questions from this chapter

Show that the polar equation of the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \quad\) is \(\quad r^2=\dfrac{b^2}{1-e^2 \cos^2\ \theta}\).

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=4(1 - \sin\ \theta)\)

An equation of the form \(r=\dfrac{ep}{1+e\cos\ \theta}\) has a ________ directrix to the ________ of the pole.

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=4 + 3 \cos\ \theta\)

In Exercises 59-64, use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. \(r=2\ \cos \left(\dfrac{3\theta}{2}\right)\)
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