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Chapter 10: Problem 102

Convert the polar equation to rectangular form. $$r=10$$

Short Answer

Expert verified
The rectangular form of the polar equation \(r=10\) is \(x^2 + y^2 = 100\)

Step by step solution

01

Understanding Polar and Rectangular Form

In the polar equation \(r = 10\), the equation specifies an area that is a fixed distance from the origin, creating a perfect circle centered at the origin. As this equation is very simple, it does not involve the angle \(\theta\), it implies that the radius has a fixed distance, which is 10, around the origin
02

Convert the Polar Equation to Rectangular Equation

In the rectangular coordinate system, a circle with radius \(r\) centered at the origin is given by \(x^2 + y^2 = r^2\). We can use this to transform our polar equation. Substituting \(r = 10\) into the equation, we get \(x^2 + y^2 = 10^2\)

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Most popular questions from this chapter

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. $$\begin{aligned}&r_{1}=\frac{4}{1+0.4 \cos \theta}\\\&r_{2}=\frac{4}{1-0.4 \sin \theta}\end{aligned}$$ (c) Use a graphing utility to verify your results in part (b).

Convert the polar equation to rectangular form. Then sketch its graph. $$\theta=\pi / 6$$

Identify the conic and sketch its graph. $$r=\frac{3}{2+4 \sin \theta}$$

Write the polar equation of the conic for \(e=1, e=0.5,\) and \(e=1.5\) Identify the conic for each equation. Verify your answers with a graphing utility. $$r=\frac{2 e}{1+e \sin \theta}$$

Convert the polar equation to rectangular form. $$r^{2}=\sin 2 \theta$$
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