The points (0,4) and (1,8) lie on a circle whose center is on the x-axis. What is the radius of the circle?
+23252 The center of the circle is on the perpendicular bisector of each chord of the circle.
The plan:
1) find the equation of the perpendicular bisector of the chord whose endpoints are (0, 4) and (1, 8),
2) find where this equation crosses the x-axis; this will be the center.
3) find the distance from this point (the center) to the point (0, 4).
1) The midpoint of (0, 4) and (1, 8) is ( ½, 6).
The slope of the chord is: m = (8 - 4) / (1 - 0) = 4.
The perpendicular will have a slope of - ¼.
The equation of this line is: y - 6 = - ¼( x - ½ ).
Multiplying by -4: -4y + 24 = x - ½
Multiplying by 2: -8y + 48 = 2x - 1
2) On the x-axis, the value of y is zero.
Making this subsitution: -8(0) + 48 = 2x - 1
49 = 2x ---> x = 24.5
So, the center is (24.5, 0).
3) Finding the distance from (24.5, 0) to (0, 4):
distance = sqrt( (4 - 0)2 + (0 - 24.5)2 ) = sqrt( 16 + 600.25 ) = sqrt( 616.25 )
+37146 Here is a second way:
the two given points are equidistant from the center (x, 0)
Use distance formulas for the two points and the center
(x-0)^2 + 4^2 = (x-1)^2 + 8^2
solve for x = 49/2 so circle center is 49/2 , 0
use one pont to solve for r in the circle formula (0,4 )
( 0 - 49/2)^2 + (4-0)^2 = r^2
r^2 = 616.25 r = sqrt (616.25)
