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Chapter 8: Problem 19

Convert the rectangular equation to polar form and sketch its graph. $$ x^{2}+y^{2}=a^{2} $$

Short Answer

Expert verified
The polar form of the given equation is \(r=|a|\). The graph is a circle with radius \(a\) if \(a>0\) and circle with radius \(-a\) from the opposite direction if \(a<0\) centered at the origin.

Step by step solution

01

Convert Rectangular Equation to Polar Form

Use the conversion relations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Substituting these into the equation results in \((r\cos(\theta))^{2}+(r\sin(\theta))^{2}=a^{2}\), which simplifies to \(r^{2}=a^{2}\). Taking square root of both sides gives \(r=|a|\), hence, the polar form is \(r=a\) for \(a>0\) and \(r=-a\) for \(a<0\).
02

Graphing the Polar Equation

The graph of \(r=a\) is a circle centered at the origin with a radius of \(a\), if \(a>0\), and likewise \(r=-a\) is also a circle centered at the origin, but is conventionally represented in polar coordinates by the negative radius under the graph \(r=a\). Therefore, if \(a<0\), then move in the opposite direction to the initial direction that theta indicates.

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Most popular questions from this chapter

Use the formula for the arc length of a curve in parametric form to derive the formula for the arc length of a polar curve.

Writing Consider the polar equation \(r=\frac{4}{1+e \sin \theta} .\) (a) Use a graphing utility to graph the equation for \(e=0.1\), \(e=0.25, e=0.5, e=0.75,\) and \(e=0.9 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{-}\) and \(e \rightarrow 0^{+}\) (b) Use a graphing utility to graph the equation for \(e=1\). Identify the conic. (c) Use a graphing utility to graph the equation for \(e=1.1\), \(e=1.5,\) and \(e=2 .\) Identify the conic and discuss the change in its shape as \(e \rightarrow 1^{+}\) and \(e \rightarrow \infty\).

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Cycloid: } x=2(\theta-\sin \theta), \quad y=2(1-\cos \theta) $$

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=a(1+\cos \theta) & 0 \leq \theta \leq \pi & \text { Polar axis } \end{array} $$

In Exercises 57 and \(58,\) let \(r_{0}\) represent the distance from the focus to the nearest vertex, and let \(r_{1}\) represent the distance from the focus to the farthest vertex. Show that the eccentricity of an ellipse can be written as \(e=\frac{r_{1}-r_{0}}{r_{1}+r_{0}} .\) Then show that \(\frac{r_{1}}{r_{0}}=\frac{1+e}{1-e}\).
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