A circle has equation (x - 2)^2 + (y + 3)^2 = 18. 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 3) squared equals 18. To find the center and radius, we need to compare this with the standard form of a circle equation: (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. The term (x minus h) squared corresponds to (x minus 2) squared, so h equals 2. The term (y minus k) squared corresponds to (y plus 3) squared, which can be written as (y minus negative 3) squared, so k equals negative 3. Therefore, the center of the circle is at the point (2, -3).
To find the radius, we look at the right side of the equation, which is 18. This represents r squared. To find r, we take the square root of both sides: r equals the square root of 18. We can simplify the square root of 18 by factoring it into 9 times 2. The square root of 9 is 3, and the square root of 2 remains as is. So, r equals 3 times the square root of 2. This is approximately 4.24 units.
Let's visualize the circle on a coordinate plane. The center is at the point (2, -3), marked in red. The radius is 3 times the square root of 2, which is approximately 4.24 units. We draw the circle with this center and radius. A yellow line shows the radius from the center to the edge of the circle.
To verify our solution, we can pick a point on the circle and check if it satisfies the original equation. Let's use the point directly to the right of the center: (2 plus 3 root 2, negative 3). Substituting into the equation: ( (2 plus 3 root 2) minus 2 ) squared plus ( (negative 3) plus 3 ) squared. This simplifies to (3 root 2) squared plus 0 squared, which is 18 plus 0, equaling 18. This confirms our center and radius are correct.
To summarize: We were given the equation (x minus 2) squared plus (y plus 3) squared equals 18. By comparing it to the standard form, we identified the center as (2, -3). The radius was found by taking the square root of 18, which simplifies to 3 root 2, or approximately 4.24 units. The visual representation confirms these findings.