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Chapter 10: Problem 94

Convert the polar equation \(r=\cos \theta+3 \sin \theta\) to rectangular form and identify the graph.

Short Answer

Expert verified
In rectangular coordinates the polar equation \(r=\cos \theta+3 \sin \theta\) becomes the equation of a circle \((x - \frac{1}{2})^2 + (y - \frac{3}{2})^2 = \frac{5}{4}\). The graph represents a circle centered at \((\frac{1}{2},\frac{3}{2})\) with a radius of \(\frac{\sqrt{5}}{2}\).

Step by step solution

01

Convert Polar to Rectangular

In polar coordinates, \(r=\cos \theta+3 \sin \theta\). To convert this into rectangular coordinates, use the relationships \(x=r\cos \theta\) and \(y=r\sin \theta\). Substitute into the given polar equation: \(r = \sqrt{x^2 + y^2} = x/r + 3y/r\). Multiply through by r, recognizing that in rectangular coordinates, \(r^2 = x^2 + y^2\). This results in \(x^2 + y^2 = x + 3y\).
02

Rearrange to a Standard Form

To identify the curve, it helps to have the equation in a standard form. Bring every term to one side of the equation: \(x^2 -x + y^2 -3y = 0 \). Complete the square for the x and y terms. This results into: \( (x - \frac{1}{2})^2 + (y - \frac{3}{2})^2 = \frac{5}{4} \). This is the equation of a circle in standard form.
03

Identify the Graph

With the equation written in standard form, the graph can be identified. The equation \( (x - h)^2 + (y - k)^2 = r^2 \) represents a circle whose center is at the point \((h, k)\) and whose radius is \(r\). Here, \(h = \frac{1}{2}\), \(k = \frac{3}{2}\), and \(r = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}\). Therefore, the equation represents a circle centered at \((\frac{1}{2}, \frac{3}{2})\) with radius \(\frac{\sqrt{5}}{2}\).

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Most popular questions from this chapter

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &\left(10, \frac{\pi}{2}\right)\end{array}$$

In your own words, define the term eccentricity and explain how it can be used to classify conics. Then explain how you can use the values of \(b\) and \(c\) to determine whether a polar equation of the form $$r=\frac{a}{b+c \sin \theta}$$ represents an ellipse, a parabola, or a hyperbola.

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Parabola} &e=1&y=-4\end{array}$$

Convert the polar equation to rectangular form. $$r=\frac{6}{2-3 \sin \theta}$$

Convert the polar equation to rectangular form. $$r=\frac{1}{1-\cos \theta}$$
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