Final Answer:
The radius of the circle with equation (x-1)^2 + (y+2)^2 = 9 is 3.
Step-by-step explanation:
To find the center and radius of a circle given its equation in standard form, we follow these steps:
1. Identify the center and radius by finding the values of x and y in the quadratic equation (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.
2. In our case, we have (x-1)^2 + (y+2)^2 = 9. This is a form of the standard equation with center at (1, -2) and unknown radius r.
3. To find the radius, we need to isolate r^2 on one side of the equation. We can do this by taking the square root of both sides:
(x-1)^2 + (y+2)^2 = 9
Square root both sides:
r = ±√(9 - (x-1)^2 - (y+2)^2)
4. Since we're looking for a positive value for the radius, we take the positive square root:
r = ±√(9 - (x-1)^2 - (y+2)^2) = ±√(9 - x^2 - y^2 - 4x - 8y)
5. The ± sign means that there are two possible values for r, one for each sign choice. However, since we're looking for the actual radius, we choose the positive sign:
r = ±√(9 - x^2 - y^2 - 4x - 8y) = +√(9 - x^2 - y^2 - 4x - 8y)
6. Therefore, the radius of our circle is given by:
r = +√(9 - x^2 - y^2 - 4x - 8y) = +√(9 - x^2 - y^2 - 4(x-1)-8(y+2)) = +√(3^2) = 3 units.