Suppose lighthouse A is located at the origin and lighthouse B is located at coordinates (0,6). The captain of a ship has determined that the ship's distance from lighthouse A is 2 and its distance from lighthouse B is 5. What are the possible coordinates for the ship?

Answer 1

Let the coordinates of the ship be P(x,y)
so AP =2 and BP=5

√(x^2+y^2) = 2
x^2+y^2 = 4 , (#1)

√( (x^2 + (y-6)^2 ) = 5
x^2 + y^2 - 12y + 36 = 25
x^2 + y^2 - 12y = -11 , (#2)

#1 - #2 :

12y = 15
y = 15/12 = 5/4 = 1.25
then in #1:
x^2 + 225/144 = 4
x^2 = 39/16
x = ± √39/4

The ship could be at (√39/4 , 5/4) or (-√39/4 , 5/4)