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Step-by-Step Solution
Step 1: Identify the center and radius of the first circle
The equation of the first circle is
x^2 + y^2 - 2x - 6y + 6 = 0 .
We complete the square for both x and y:
x^2 - 2x + y^2 - 6y + 6 = 0.
Rewriting:
(x^2 - 2x + 1) + (y^2 - 6y + 9) = 1 + 9 - 6.
(x - 1)^2 + (y - 3)^2 = 4.
Therefore, the center of this circle is O_1(1, 3) and its radius is
r_1 = 2 .
Step 2: Note the center of the second circle
The second circle, denoted by 'C', has its center at O_2(2, 1) and we are to find its radius, say r_2 .
Step 3: Recognize the diameter relationship
We are told that one of the diameters of the first circle is a chord of the second circle. A diameter of the first circle has endpoints lying on the first circle, and these same endpoints must lie on the second circle as well.
Step 4: Use the distance formula between centers
The distance between the two centers O_1(1, 3) and O_2(2, 1) is:
O_1O_2 = \sqrt{(2 - 1)^2 + (1 - 3)^2} = \sqrt{1 + 4} = \sqrt{5}.
Step 5: Apply the key relationship for a diameter being a chord
If the diameter of length 2r_1 from the first circle is a chord of the second circle, then by geometry of circles (specifically using right triangles formed when a chord is drawn from the first circle's diameter to the second circle's center), we have:
O_1O_2^2 + r_1^2 = r_2^2.
Substituting the known values:
(\sqrt{5})^2 + (2)^2 = r_2^2 \quad \Longrightarrow \quad 5 + 4 = r_2^2.
Therefore,
r_2^2 = 9 \quad \Longrightarrow \quad r_2 = 3.
Step 6: State the final answer
Thus, the radius of the second circle is 3.
