A circle has equation (x - 3)^2 + (y + 3)^2 = 19. 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 3) squared equals 19. The standard form of a circle equation is (x minus h) squared plus (y minus k) squared equals r squared, where (h, k) is 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 h equals x minus 3, so h equals 3. For the y-term: y minus k equals y plus 3, which is the same as y minus negative 3, so k equals negative 3. Therefore, the center is at point (3, negative 3).
To find the radius, we look at the right side of the equation: r squared equals 19. Taking the square root of both sides gives us r equals the square root of 19. This is the exact form. The approximate decimal value is 4.36.
Let's visualize this on a coordinate plane. We plot the center at point (3, negative 3). The circle has a radius of square root of 19, which is approximately 4.36. I've drawn one radius line to show the distance from the center to the circle.
To verify our solution, let's check that a point on the circle satisfies the original equation. We'll use the point (3, negative 3 plus square root of 19). Substituting into the equation: (3 minus 3) squared plus ((negative 3 plus square root of 19) plus 3) squared. This simplifies to 0 squared plus (square root of 19) squared, which equals 0 plus 19, giving us 19. This matches the right side of our equation, confirming our solution is correct.
To summarize: The center of the circle is at point (3, negative 3) and the radius is the square root of 19, which is approximately 4.36. The key steps we followed were: first, identify the standard form of the circle equation; second, compare the coefficients to find the center coordinates, being careful with signs; and third, take the square root of the constant term to find the radius.