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Chapter 11: Problem 39

Tabulate and plot enough points to sketch a graph of the following equations. $$r(\sin \theta-2 \cos \theta)=0$$

Short Answer

Expert verified
A short answer based on the solution: To sketch the graph of the polar equation \(r(\sin\theta - 2 \cos\theta) = 0\), we first converted it to rectangular coordinates, resulting in the equation \(y - 2x = 0\). We then found the x and y-intercepts, which were both at the origin, and calculated another point on the line by setting \(x=2\) and finding that \(y=4\). We then plotted these points on the coordinate plane and drew a line through them, giving us the graph of the equation in rectangular coordinates.

Step by step solution

01

Convert the equation into rectangular coordinates

To convert the given equation into rectangular coordinates, we will replace \(r\sin\theta\) by \(y\) and \(r\cos\theta\) by \(x\). The given equation is: $$r(\sin\theta - 2 \cos\theta) = 0$$ Multiplying through by r, we get: $$y - 2x = 0$$
02

Find the range of values for x and y

Since this equation represents a straight line, we need to find at least two points to sketch the graph. We will find the x and y-intercepts for the line. To find the x-intercept, set y to 0, and to find the y-intercept, set x to 0.
03

Tabulate a list of points

To find the x-intercept, let \(y=0\) and solve for x: $$0 - 2x = 0$$ $$x = 0$$ To find the y-intercept, let \(x=0\) and solve for y: $$y - 2(0) = 0$$ $$y = 0$$ We've found that the x-intercept and y-intercept are both 0. This means that the line passes through the origin. To get another point on the line, let's arbitrarily choose \(x=2\): $$y - 2(2) = 0$$ $$y = 4$$ So, when \(x=2\), \(y=4\). Now we have two points to plot: \((0,0)\) and \((2,4)\).
04

Plot the points and sketch the graph

Plot the points (0,0) and (2,4) on the coordinate plane and draw the line that passes through them. This is the graph of the equation \(y-2x=0\) in rectangular coordinates, which corresponds to the polar equation \(r(\sin\theta-2\cos\theta)=0\).

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Most popular questions from this chapter

Modify Figure 56 to derive the polar equation of a conic section with a focus at the origin in the following three cases. a. Vertical directrix at \(x=-d,\) where \(d > 0\) b. Horizontal directrix at \(y=d,\) where \(d > 0\) c. Horizontal directrix at \(y=-d,\) where \(d > 0\)

Use a graphing utility to graph the parabolas \(y^{2}=4 p x,\) for \(p=-5,-2,-1,1,2,\) and 5 on the same set of axes. Explain how the shapes of the curves vary as \(p\) changes.

Prove that the equations $$x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t$$ where \(a, b, c,\) and \(d\) are real numbers, describe a circle of radius \(R\) provided \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and \(a b+c d=0\)

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{4}{2+\cos \theta}$$

Give the property that defines all hyperbolas.
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