Posted by Anonymous on Thursday, February 14, 2013 at 11:24am.
If the line touches the circle, then the line from (2,5) to the center is perpendicular to the line y=2x+1, so it has slope -1/2.
So, the line through (2,5) with slope -1/2 is
y-5 = -1/2 (x-2)
y = -1/2 x + 6
This line passes through the center of the circle, so
-1/2 x + 6 = 9-x
x=6
(6,3) is the center of the circle.
The radius is the distance from (6,3) to (2,5) = √20
The circle is thus
(x-6)^2 + (y-3)^2 = 20
Just as a final check:
The distance from (6,3) to the line 2x-y+1 = 0 is |2*6 - 3 + 1|/√5 = 10/√5 = 2√5 = √20
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