There's surely an easier way to do #5 . . . but here's my solution.
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The center of the circle is .
Since the circle is tangent to the line at point ,
. . the radius at is perpendicular to the line.
The center lies along this perpendicular.
The line is: . with slope .
The equation of the perpendicular is: . 
The center also lies on the perpendicular bisector of chord .
The midpoint of is:
The slope of
. . The slope of the perpendicular bisector is:
The equation of the bisector is: . 
The center is the intersection of  and : .
. . Hence, the center is:
The radius is the length of
Therefore, the equation of the circle is: .