Question

What are the coordinates of the center of the circle that passes through the points (1, 1), (1, 5), and (5, 5)?

Answer 1

Let the location of the center (x,y)

let the distance between center and point (5,5) = d₁

let the distance between center and point (1,5) = d₂

let the distance between center and point (1,1) = d₃

 

The condition of the circle is d₁ = d₂ = d₃

Making two equations to find x , y

 

d₁² = d₂²

∴ (x-5)² + (y-5)² = (x-1)² + (y-5)²

∴ (x-5)²  = (x-1)²

∴ x-5 = \pm (x-1) ⇒⇒⇒ positive sign unacceptable

∴ x-5 = -(x-1)

∴ x = 3

 

d₁² = d₃²

∴ (x-5)² + (y-5)² = (x-1)² + (y-1)²

substituting with x and simplifying the equation

∴ y = 3

 

∴center = (3,3)

 The correct choice will be (C)

Answer ⇒⇒⇒ C. (3, 3)