Answer:
(x +2.5)² +(y -2)² = 6.25
Explanation:
A radius of a circle is perpendicular to the tangent line at the point of tangency. This means the center of the circle lies on the horizontal line through the point (0, 2), which is on the vertical tangent x=0 (the y-axis).
The center of a circle lies on the perpendicular bisector of any chord, so the perpendicular bisector of the given two points will pass through the circle center.
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equation of the chord bisector
The slope of the chord between the given points is found using the slope formula:
m = (y2 -y1)/(x2 -x1)
m = (2 -0)/(0 -(-1)) = 2 ⇒ perpendicular has slope -1/2
The midpoint of the chord is ...
M = (A +B)/2 = ((0, 2) +(-1, 0))/2 = (-1, 2)/2 = (-1/2, 1)
The point-slope form of the perpendicular through the midpoint of the chord is ...
y -1 = -1/2(x -(-1/2))
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center of the circle
For y = 2, the x-value is ...
2 -1 = -1/2(x +1/2)
-2 = x +1/2 . . . . . . multiply by -2
-2.5 = x . . . . . . . subtract 1/2
The circle center is (-2.5, 2), and the radius is 2.5 (distance from center to y-axis).
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equation of the circle
The equation can be written as ...
(x -h)² +(y -k)² = r² . . . . . . circle with center (h, k) and radius r
(x -(-2.5))² +(y -2)² = 2.5²
(x +2.5)² +(y -2)² = 6.25