Write the standard form of the equation of the circle with the given center and radius. Center (−3, −1), r = √3

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Recall that the standard form of the equation of a circle with center $(h, k)$ and radius $r$ is given by the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \]

Identify the center $(h, k)$ and radius $r$ from the problem. Here, the center is $(-3, -1)$ and the radius is $\sqrt{3}$.

Substitute the values of $h = -3$, $k = -1$, and $r = \sqrt{3}$ into the standard form equation: \[ (x - (-3))^2 + (y - (-1))^2 = (\sqrt{3})^2 \]

Simplify the expressions inside the parentheses by removing the double negatives: \[ (x + 3)^2 + (y + 1)^2 = (\sqrt{3})^2 \]

Square the radius $\sqrt{3}$ to get $3$, so the equation becomes: \[ (x + 3)^2 + (y + 1)^2 = 3 \]

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form directly relates the coordinates of any point on the circle to its center and radius.

Coordinates of the Center

The center of the circle is given as a point (h, k). In this problem, the center is (-3, -1), which means the equation will use (x + 3) and (y + 1) to represent the horizontal and vertical distances from the center.

Radius and Its Square

The radius r is the distance from the center to any point on the circle. Since the equation uses r², you must square the given radius. Here, r = √3, so r² = 3, which will be the constant on the right side of the equation.

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