What are the coordinates of the center of the circle?

Concept:

The distance of the center from any point present on the circumference of the circle is always same. (which is also called the radius of the circle)

The distance between two points (a, b) and (c, d) is defined as

√[(a - c)² + (b - d)²]

Calculation:

Let O(x, y) be the center of the circle.

Accordin to the concept used

OA = OB = OC 

⇒ OA2 = OB2  = OC2 

So, By taking OA² = OB² , we get 

⇒ (x − 5)² + (y + 8)² =  (x + 2)² + (y - 9)²

⇒ x² + 25 - 10x + y² + 64 + 16y = x² + 4 + 4x + y² + 81 - 18y

⇒ 89 - 10x + 16y = 85 + 4x - 18y

⇒ 89 - 85 - 10x + 16y - 4x + 18y = 0

⇒ 4 - 14x + 34y = 0 

⇒ 14x - 34y = 4      ----(1)        

And By taking OA2 = OC2 , we get 

⇒ (x − 5)2 + (y + 8)2 = (x - 2)² + (y - 1)²

⇒ x² + 25 - 10x + y² + 64 + 16y = x² + 4 - 4x + y² + 1 - 2y

⇒ 89 - 10x + 16y = 5 - 4x - 2y

⇒ 89 - 5 - 10x + 4x + 16y + 2y = 0

⇒ 84 - 6x + 18y = 0 

⇒ 6x - 18y = 84      ----(2)    

On solving (1) and (2), we get

y = -24 and x = -58

∴ The coordinates of the center of the circle is (-58, - 24).