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Chapter 6: Problem 74

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

Short Answer

Expert verified
The polar equation translates into the rectangular form \(0 = 3x + 4y^2\). This equation represents a parabola with its axis of symmetry parallel to the x-axis, opening towards the positive x-axis.

Step by step solution

01

Convert the polar equation to rectangular form

Start by restating the given polar equation in terms of \(x\) and \(y\) using the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). The equation can be rewritten as \(r = x + 3y\). As \(r\) is also equal to \(\sqrt{x^2 + y^2}\), we can replace \(r\) with \(\sqrt{x^2 + y^2}\), giving \(\sqrt{x^2 + y^2} = x + 3y\).
02

Simplify the Rectangular Equation

Now, by squaring both sides of the equation to eliminate the square root, we have \(x^2 + y^2 = x^2 + 6xy + 9y^2\). Subtract \(x^2\) and \(9y^2\) from both sides to simplify further to \(0 = 6xy + 8y^2\). Divide all terms by 2, yielding \(0 = 3x + 4y^2\).
03

Identify the graph

The resulting equation after simplification, \(0 = 3x + 4y^2\), is the equation for a parabola with its axis of symmetry parallel to the x-axis. The direction of the opening of the parabola is determined by the sign of the term that multiplies \(y^2\). Here, the coefficient of \(y^2\) is positive, indicating that the opening of the parabola is towards the positive x-axis.

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Most popular questions from this chapter

\(r=\frac{6}{2 \sin \theta-3 \cos \theta}\)

In Exercises 29-32, use a graphing utility to graph the rotated conic. $$ r=\frac{3}{3+\sin (\theta-\pi / 3)} $$

The comet Encke has an elliptical orbit with an eccentricity of \(e \approx 0.847\). The length of the major axis of the orbit is approximately \(4.42\) astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun?

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (10, \pi / 2) \\ \end{array} $$

Consider the polar equation $$ r=\frac{4}{1-0.4 \cos \theta} $$ (a) Identify the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. $$ r_{1}=\frac{4}{1+0.4 \cos \theta}, \quad r_{2}=\frac{4}{1-0.4 \sin \theta} $$ (c) Use a graphing utility to verify your results in part (b).
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