The equation of a circle is (x^{2}-4x+3)+(y^{2}-6y+8)=0. Which of the following statements are correct? I. The end points of a diameter of the circle are at (1, 2) and (3, 4). II. The end points of a diameter of the circle are at (1, 4) and (3, 2). III. The end points of a diameter of the circle are at (2, 4) and (4, 2). Select the answer using the code given below.

Explanation

The equation can be factored as (x-1)(x-3) + (y-2)(y-4) = 0. This is the diameter form of a circle (x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0. Thus, (1,2) and (3,4) are endpoints of a diameter, confirming Statement I. The equation can also be written as (x-1)(x-3) + (y-4)(y-2) = 0, meaning (1,4) and (3,2) are also endpoints of a diameter, confirming Statement II. The center is (2,3), but the midpoint of (2,4) and (4,2) is (3,3), so Statement III is incorrect.

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