EQUATION OF A CIRCLE WHEN EXTREMITIES OF A DIAMETER ARE GIVEN

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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