Question

80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.r² - 8r cos(θ - π/2) = 9

Accepted Answer

Recognize that the given equation is in polar form: $$r^{2} - 8r$$ \cos(	heta - \frac{\pi}{2}) = 9$$. Our goal is to rewrite this in a more recognizable form, such as the standard form of a circle in Cartesian coordinates. Recall the polar to Cartesian coordinate conversions: x = r$$ \cos 	heta$$ and y = r$$ \sin 	heta$$. Also, note that $$\cos(	heta - \alpha) = \cos 	heta \cos \alpha + \sin 	heta \sin \alpha$$. Apply the cosine difference identity to expand $$\cos(	heta - \frac{\pi}{2})$$:

$$\cos\left(	heta - \frac{\pi}{2}ight) = \cos 	heta \cos \frac{\pi}{2} + \sin 	heta \sin \frac{\pi}{2} = 0 + \sin 	heta = \sin 	heta$$. Substitute this back into the original equation to get:

r$$^{2}$$ - 8r \sin 	heta = 9.

$$Then, rewrite $$r^{2} as$$ $$x^{2}$$ + $$y^{2}$$ and $$r \sin 	heta as$$ $$y$$, giving:

$$x^{2}$$ + $$y^{2} - 8y = 9$. Complete the square for the y$ terms to write the equation in standard circle form:

x$$^{2}$$ + (y$$^{2}$$ - 8y + 16) = 9 + 16$$

which simplifies to:

$$x^{2} + (y$$ - $$4)^{2} = 25$.

This represents a circle centered at (0$$, 4)$$ with radius 5$.

80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.

r² - 8r cos(θ - π/2) = 9

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Key Concepts

Polar Coordinates and Equations

Conversion Between Polar and Cartesian Forms

Standard Equation of a Circle in Polar Coordinates