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The coordinates of a point A, where AB is the diameter of a circle whose centre is (2, тАУ 3) and B is (1, 4) is:
(тИТ2,10)
(2,тИТ10)
(3,тИТ10)
(тИТ3,10)
(3,тИТ10)
Use the symmetry of the midpoint: Center x = (AxтАЛ + BxтАЛ)/2. Since 2 = (x + 1)/2, then x + 1 = 4, so x = 3. Only option C matches x = 3.
The center of the circle is (2, -3), which is the midpoint of the diameter AB, and the coordinates of point B are (1, 4).
(2x1тАЛ+x2тАЛтАЛ,2y1тАЛ+y2тАЛтАЛ)=(x,y)
Use the symmetry of the midpoint: Center x = (AxтАЛ + BxтАЛ)/2. Since 2 = (x + 1)/2, then x + 1 = 4, so x = 3. Only option C matches x = 3.
Students often mistakenly use the distance formula or assume the center is the endpoint instead of the midpoint of the diameter.
Define Midpoint Coordinates
Let the coordinates of point A be (x,y). Since the center (2,тИТ3) is the midpoint of the diameter AB with B(1,4), we apply the midpoint formula.
Midpoint=(2x+1тАЛ,2y+4тАЛ)=(2,тИТ3)
Solve for the x-coordinate
Equate the x-coordinates to solve for x: 2x+1тАЛ=2. Multiplying by 2 gives x+1=4, so x=3.
x=3
Solve for the y-coordinate
Equate the y-coordinates to solve for y: 2y+4тАЛ=тИТ3. Multiplying by 2 gives y+4=тИТ6, so y=тИТ10.
y=тИТ10
C is correct because calculating the midpoint coordinates yields the point (3,тИТ10).
This concept is foundational for Coordinate Geometry; remember that any line segment passing through the center of a circle connecting two points on the circumference is always bisected by the center.
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