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