The circle passing through the point (5,0) and touching the y-axis at (0,2) also passes through the point

Since the circle touches the y-axis at (0,2), its center will be (h,2) and radius will be h. Equation of circle is given by
(x-h)² + (y-2)² = h²
Since the point (5,0) lies on the circle,
(5-h)² + (0-2)² = h²
25 - 10h + h² + 4 = h²
29 - 10h = 0
10h = 29
h = 2.9
Therefore, the equation of the circle is
(x-2.9)² + (y-2)² = (2.9)²
Let's check the given options:
(8,0): (8-2.9)² + (0-2)² = 26.01 + 4 = 30.01 ≠ (2.9)²
(8,52): (8-2.9)² + (52-2)² = 26.01 + 2500 = 2526.01 ≠ (2.9)²
(12,2): (12-2.9)² + (2-2)² = 82.81 ≠ (2.9)²
(8,0): (8-2.9)² + (0-2)² = 26.01 + 4 = 30.01 ≠ (2.9)²
Let's find the equation of the circle again.
The center is (r,2) and radius is r.
The equation of the circle is (x-r)² + (y-2)² = r²
Since it passes through (5,0),
(5-r)² + (0-2)² = r²
25 - 10r + r² + 4 = r²
29 - 10r = 0
r = 2.9
The equation of the circle is (x-2.9)² + (y-2)² = (2.9)² = 8.41
Let's check the options:
A. (8,0): (8-2.9)² + (0-2)² = 26.01 + 4 = 30.01 ≠ 8.41
B. (8,52): (8-2.9)² + (52-2)² = 26.01 + 2500 = 2526.01 ≠ 8.41
C. (12,2): (12-2.9)² + (2-2)² = 82.81 ≠ 8.41
D. (8,0): (8-2.9)² + (0-2)² = 26.01 + 4 = 30.01 ≠ 8.41
None of the options satisfy the equation of the circle.