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

검증된 단계별 안내

Recognize that the given equation is in polar form: \(r^{2} - 8r \cos(\theta - \frac{\pi}{2}) = 9\). 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: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, note that \(\cos(\theta - \alpha) = \cos \theta \cos \alpha + \sin \theta \sin \alpha\).

Apply the cosine difference identity to expand \(\cos(\theta - \frac{\pi}{2})\): \(\cos\left(\theta - \frac{\pi}{2}\right) = \cos \theta \cos \frac{\pi}{2} + \sin \theta \sin \frac{\pi}{2} = 0 + \sin \theta = \sin \theta\).

Substitute this back into the original equation to get: \(r^{2} - 8r \sin \theta = 9\). Then, rewrite \(r^{2}\) as \(x^{2} + y^{2}\) and \(r \sin \theta\) 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 \(5\).

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주요 개념

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Polar Coordinates and Equations

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). Understanding how to interpret and manipulate equations in polar form is essential for analyzing curves like circles defined by r and θ.

Conversion Between Polar and Cartesian Forms

Converting polar equations to Cartesian form (using x = r cos θ and y = r sin θ) helps visualize and graph curves. This conversion is crucial for recognizing standard geometric shapes, such as circles, from polar equations.

Standard Equation of a Circle in Polar Coordinates

A circle in polar coordinates can often be expressed as r = 2a cos(θ - α) + b or similar forms. Recognizing these forms and relating them to the circle's center and radius aids in graphing and understanding the geometric properties of the circle.

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