Circles

Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.

r = −2 cos θ

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Recall that the general form of a circle in polar coordinates can often be written as \(r = a \cos \theta + b \sin \theta + c\). In this problem, the equation is \(r = -2 \cos \theta\), which can be analyzed to find the center and radius of the circle.

Rewrite the equation \(r = -2 \cos \theta\) by factoring out the negative sign: \(r = - (2 \cos \theta)\). This suggests the circle is related to the cosine function with a negative coefficient.

Convert the polar equation to Cartesian coordinates using the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\). Multiply both sides of the equation by \(r\) to get \(r^2 = -2 r \cos \theta\).

Substitute \(r^2 = x^2 + y^2\) and \(r \cos \theta = x\) into the equation to obtain \(x^2 + y^2 = -2x\). Rearrange this to \(x^2 + 2x + y^2 = 0\).

Complete the square for the \(x\)-terms: \(x^2 + 2x + 1 + y^2 = 1\). This can be written as \((x + 1)^2 + y^2 = 1\), which represents a circle with center at \((-1, 0)\) in Cartesian coordinates and radius \(1\). Convert the center to polar coordinates by calculating \(r = \sqrt{(-1)^2 + 0^2}\) and \(\theta = \arctan(0 / -1)\).

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주요 개념

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Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle (r, θ) from the origin. Unlike Cartesian coordinates, the position depends on the distance from the origin and the direction, making it useful for curves like circles defined by r as a function of θ.

Equation of a Circle in Polar Form

A circle in polar coordinates can often be expressed as r = a ± b cos θ or r = a ± b sin θ. These forms correspond to circles with centers offset from the origin, and understanding how to convert or interpret these equations helps identify the circle's center and radius.

Conversion Between Polar and Cartesian Coordinates

To analyze or sketch polar curves, converting between polar (r, θ) and Cartesian (x, y) coordinates is essential. Using x = r cos θ and y = r sin θ allows one to rewrite the polar equation in Cartesian form, facilitating identification of geometric properties like the center and radius of a circle.

CirclesSketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.r = −2 cos θ