calculus

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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?

  • calculus -

    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)

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