Question

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

A. (0, 0)
B. (5, 1)
C. (3, 3)
D. (2.5, 2.5)

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)