Answer: The general equation for a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
To find the equation of the circle that passes through the points (1, 1), (1, 3), and (9, 2), we can use the fact that the perpendicular bisectors of any two chords of a circle intersect at the center of the circle.
The midpoint and slope of the chord connecting (1, 1) and (1, 3) are:
Midpoint: ((1+1)/2, (1+3)/2) = (1, 2)
Slope: undefined (since x values are the same)
The perpendicular bisector of this chord passes through the midpoint (1, 2) and has a slope of 0. Therefore, its equation is:
y - 2 = 0
Simplifying this equation gives:
y = 2
Similarly, the midpoint and slope of the chord connecting (1, 1) and (9, 2) are:
Midpoint: ((1+9)/2, (1+2)/2) = (5, 1.5)
Slope: (2 - 1)/(9 - 1) = 1/8
The perpendicular bisector of this chord passes through the midpoint (5, 1.5) and has a slope of -8 (the negative reciprocal of the slope of the chord). Therefore, its equation is:
y - 1.5 = -8(x - 5)
Simplifying this equation gives:
y = -8x + 41.5
The intersection of the two perpendicular bisectors (y = 2 and y = -8x + 41.5) is the center of the circle. Solving for x gives:
2 = -8x + 41.5
8x = 39.5
x = 4.9375
Substituting x = 4.9375 into either equation for the perpendicular bisectors gives:
y = 2
Therefore, the center of the circle is (4.9375, 2) and the radius is the distance from the center to any of the three given points. For example, the distance from (4.9375, 2) to (1, 1) is:
r = sqrt((4.9375 - 1)^2 + (2 - 1)^2) = sqrt(24.9219) = 4.992
So the equation of the circle is:
(x - 4.9375)^2 + (y - 2)^2 = (4.992)^2
Expanding and simplifying this equation gives:
x^2 - 9.875x + 24.526 + y^2 - 4y + 0.076 = 24.916
Rearranging terms gives:
x^2 + y^2 - 9.875x - 4y - 0.34 = 0
So the general equation for the circle is:
x^2 + y^2 - 9.875x - 4y - 0.34 = 0
Explanation: