A circle has equation (x - 2)^2 + (y + 2)^2 = 17. What is the center and radius
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We are given the equation of a circle: (x minus 2) squared plus (y plus 2) squared equals 17. The standard form of a circle equation is (x minus h) squared plus (y minus k) squared equals r squared, where h and k represent the center coordinates, and r represents the radius.
To find the center, we compare the given equation with the standard form. For the x-term: x minus h equals x minus 2, so h equals 2. For the y-term: y plus 2 equals y minus negative 2, so k equals negative 2. Therefore, the center of the circle is at point (2, negative 2).
To find the radius, we look at the right side of the equation where r squared equals 17. Taking the square root of both sides gives us r equals the square root of 17. The decimal approximation of the square root of 17 is approximately 4.12.
Let's visualize the circle on a coordinate plane. The center is at point (2, negative 2). The radius is the square root of 17, which is approximately 4.12. I've drawn the circle with this center and radius, and shown the radius as a line segment from the center to the circle.
To verify our solution, let's check if a point on the circle satisfies the original equation. I'll use the point directly above the center: (2, negative 2 plus square root of 17). Substituting into the equation: (2 minus 2) squared plus ((negative 2 plus square root of 17) plus 2) squared. This simplifies to 0 squared plus (square root of 17) squared, which equals 0 plus 17, or 17. Since this matches the right side of our equation, our center and radius are correct.
Let's summarize our findings. We were given the equation (x minus 2) squared plus (y plus 2) squared equals 17. By comparing with the standard form of a circle equation, we determined that the center is at point (2, negative 2) and the radius is the square root of 17, which is approximately 4.12. The method is to compare the given equation with the standard form (x minus h) squared plus (y minus k) squared equals r squared to find the center coordinates (h, k) and radius r equals the square root of the constant term.