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Chapter 7: Problem 78

Show that each statement is true by converting the given polar equation to a rectangular equation. Show that the graph of \(r=a \cos \theta\) is a circle with center at \(\left(\frac{a}{2}, 0\right)\) and radius \(\frac{a}{2}\)

Short Answer

Expert verified
The graph of the polar equation \(r=a \cos \theta\) corresponds to a circle in rectangular coordinates with center \((a/2, 0)\) and radius \(a/2\).

Step by step solution

01

Conversion from Polar to Rectangular Coordinates

Start with the given polar equation \(r=a \cos \theta\). Multiply both sides by \(r\) to get \(r^2 = ar \cos \theta\). Now, convert \(r^2\) and \(r \cos \theta\) to rectangular form using the relationships \(r^2 = x^2 + y^2\) and \(r \cos \theta = x\). This yields the equation \(x^2 + y^2 = ax\).
02

Reformatting the Equation

Rearranging the above equation by moving \(ax\) to the left-hand side, we get \(x^2 - ax + y^2 = 0\). The goal is to complete the square for the \(x\) terms, which means adding \((a/2)^2\) to both sides of the equation to balance it out. The equation becomes \(x^2 - ax + (a/2)^2 + y^2 = (a/2)^2\). Simplify this to get \((x - a/2)^2 + y^2 = (a/2)^2\), which is the standard form of a circle's equation.
03

Verification of Circle's Center and Radius

Comparing the equation \((x - a/2)^2 + y^2 = (a/2)^2\) with the standard form \((x-h)^2 + (y-k)^2 = r^2\), it can be seen that the circle is centered at \((h, k) = (a/2, 0)\) and has a radius of \(r = a/2\), thus validating the given statement.

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Most popular questions from this chapter

Use a graphing utility to graph the polar equation. $$r=2+4 \sin \theta$$

Find the smallest interval for \(\theta\) starting with \(\theta \min =0\) so that your graphing utility graphs the given polar equation exactly once without retracing any portion of it. $$r=4 \sin \theta$$

In Exercises \(77-80,\) convert to polar form and then perform the indicated operations. Express answers in polar and rectangular form. $$ \frac{(-1+i \sqrt{3})(2-2 i \sqrt{3})}{4 \sqrt{3}-4 i} $$

In Exercises \(69-76,\) find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth. The complex fourth roots of \(81\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)\)

Test for symmetry and then graph each polar equation. $$r=\frac{2}{1-\cos \theta}$$
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