Determine the equation of the circle with center (5, 5) containing the point (0, -2)

To determine the equation of the circle, start with the standard form of the equation of a circle:

[

(x – h)^2 + (y – k)^2 = r^2

]

where ((h, k)) is the center of the circle and (r) is the radius.

Given the center ((5, 5)):

– (h = 5)

– (k = 5)

Now, we need to find the radius (r). The radius can be determined using the distance formula between the center ((5, 5)) and the point ((0, -2)):

[

r = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} = sqrt{(0 – 5)^2 + (-2 – 5)^2}

]

Calculating this step-by-step:

1. ((0 – 5)^2 = (-5)^2 = 25)
2. ((-2 – 5)^2 = (-7)^2 = 49)
3. Add these: (25 + 49 = 74)

So,

[

r = sqrt{74}

]

Now, substitute (h), (k), and (r^2) back into the equation:

[

(x – 5)^2 + (y – 5)^