A circle has equation (x - 3)^2 + (y + 1)^2 = 20. What is the center and radius

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We are given the equation of a circle: (x minus 3) squared plus (y plus 1) squared equals 20. The standard form of a circle equation is (x minus h) squared plus (y minus k) squared equals r squared, where h and k are the coordinates of the center, and r is the radius. To find the center, we compare the given equation with the standard form. For the x-term, (x minus 3) squared corresponds to (x minus h) squared, so h equals 3. For the y-term, (y plus 1) squared can be written as (y minus negative 1) squared, which corresponds to (y minus k) squared, so k equals negative 1. Therefore, the center of the circle is at the point (3, negative 1). To find the radius, we look at the right side of the equation, which is 20. This represents r squared. To find r, we take the square root of 20. We can simplify the square root of 20 by factoring it as the square root of 4 times 5. This equals the square root of 4 times the square root of 5, which is 2 times the square root of 5. The approximate decimal value is 4.47. Here is the visual representation of our circle. The center is plotted at the point (3, negative 1). The radius is 2 times the square root of 5, which is approximately 4.47 units. The circle extends this distance in all directions from the center. To summarize, the center of the circle is at (3, negative 1) and the radius is 2 times the square root of 5. Let's verify this by substituting a point on the circle back into the original equation. We'll use the point (3 plus 2 root 5, negative 1). Substituting into (x minus 3) squared plus (y plus 1) squared, we get: (3 plus 2 root 5 minus 3) squared plus (negative 1 plus 1) squared. This simplifies to (2 root 5) squared plus zero squared, which is 4 times 5 plus 0, equaling 20. This confirms our solution is correct. The key steps were: identifying the standard form, extracting the center coordinates, and calculating the radius from the r squared value.