Equation of a circle when the extremities of the diameter are given.
(x - x1)(x - x2) + (y - y1)(y - y2) = 0
Here (x1, y1) and (x2, y2) are the extremities or end points of the diameter.
Example 1 :
Find the equation of the circle if (2, -3) and (3, 1) are the extremities of a diameter.
Solution :
Here (x1, y1) ==> (2, -3) and (x2, y2) ==> (3, 1)
(x - x1) (x - x2) + (y - y1) (y - y2) = 0
(x - 2) (x - 3) + (y - (-3)) (y - 1) = 0
(x - 2) (x - 3) + (y + 3) (y - 1) = 0
x2 - 3x - 2x + 6 + y2 - y + 3y - 3 = 0
x2 - 5x + y2 + 2y + 6 - 3 = 0
x2 + y2 - 5x + 2y + 3 = 0
Example 2 :
Find the equation of the circle described on the line joining the points (1, 2) and (2, 4) as its diameter.
Solution :
Here (x1, y1) ==> (1, 2) and (x2, y2) ==> (2, 4)
(x - x1) (x - x2) + (y - y1) (y - y2) = 0
(x - 1) (x - 2) + (y - 2) (y - 4) = 0
x2 - 2x - x + 2 + y2 - 4y - 2y + 8 = 0
x2 - 3x + 2 + y2 - 6y + 8 = 0
x2 + y2 - 3x - 2y + 2 + 8 = 0
x2 + y2 - 3x - 2y + 10 = 0
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