Question 421278
<pre><font face = "consolas" color = "indigo" size = 4><b>
Let's place the buoys at the points (-1.5,0) and (1.5,0)
so they will be 3 units (miles) apart:

{{{drawing(600,400,-3.5,5.5,-3,3, 
locate(-2.5,.3,"(-1.5,0)"), locate(1.5,.3,"(1.5,0)"),
graph(600,400,-3.5,5.5,-3,3), 
circle(1.5,0,.1), circle(-1.5,0,.1)

 )}}}

Let's draw an arbitrary variable point where the boat might be
at one instant and label it (x,y) that appears to be about twice 
as far from (-1.5,0) as it is from (1.5,0), and draw a line from 
this arbitrary variable point to each of the buoy points (-1.5,0)
and (1.5,0). Label the longer line d<sub>1</sub> and the shorter 
one d<sub>2</sub>.

{{{drawing(600,400,-3.5,5.5,-3,3, locate(1.3,1.7,"(x,y)"),
locate(-2.5,.3,"(-1.5,0)"), locate(1.5,.3,"(1.5,0)"),
graph(600,400,-3.5,5.5,-3,3), circle(1.2,1.519868415,.1),
circle(1.5,0,.1), circle(-1.5,0,.1), locate(1.4,1.1,d[2]),
green(line(1.2,1.519868415,-1.5,0),line(1.2,1.519868415,1.5,0)),
locate(-.6,1.1,d[1])

 )}}}

We set d<sub>1</sub> equal to 2 times d<sub>2</sub>

{{{d[1]=2d[2]}}}

Now we use the distance formula to substitute for the
two distances d<sub>1</sub> and d<sub>2</sub>


{{{sqrt((x+1.5)^2+(y-0)^2) = 2sqrt((x-1.5)^2+(y-0)^2)}}}

{{{sqrt((x+1.5)^2+y^2) = 2sqrt((x-1.5)^2+y^2)}}}


Square both sides:

{{{(sqrt((x+1.5)^2+y^2))^2 = (2sqrt((x-1.5)^2+y^2))^2}}}

{{{(x+1.5)^2+y^2 = 2^2(sqrt((x-1.5)^2+y^2))^2}}}

{{{(x+1.5)^2+y^2 = 4((x-1.5)^2+y^2))}}}

{{{(x+1.5)(x+1.5)+y^2 = 4((x-1.5)(x-1.5)+y^2))}}}

{{{x^2+1.5x+1.5x+2.25+y^2 = 4(x^2-1.5x-1.5x+2.25+y^2)}}}

{{{x^2+3x+2.25+y^2 = 4(x^2-3x+2.25+y^2)}}}

{{{x^2+3x+2.25+y^2 = 4x^2-12x+9+4y^2)}}}

{{{0 = 3x^2-15x+6.75+3y^2)}}}

{{{3x^2-15x+6.75+3y^2=0)}}}

Divide through by 3

x² - 5x + 2.25 + y² = 0

We can tell this is a circle.  We need to get it in
the standard form (x-h)² + (y-k)² = r²
Get the constant on the right:

x² - 5x + y² = -2.25

Complete the square on the x-terms.

Multiply the coefficient of x, which is -5 by {{{1/2}}}, 
getting -2.5.  Square 2.5, getting 6.25

Add that to both sides:

x² - 5x + y² = -2.25
 

x² - 5x + 6.25 + y² = -2.25 + 6.25

(x - 2.5)(x - 2.5) + y² = 4

Write the (x - 25)(x - 2.5) as (x - 2.5)² 
write y² as (y - 0)²
Write 4 as 2²

(x - 2.5)² + (y - 0)² = 2²

So the boat travels in a circle with center (2.5,0)
and radius 2 

Here is the circle the boat travels in:

{{{drawing(600,400,-3.5,5.5,-3,3, locate(1.3,1.7,"(x,y)"),
locate(-2.5,.3,"(-1.5,0)"), locate(1.5,.3,"(1.5,0)"),
graph(600,400,-3.5,5.5,-3,3), circle(1.2,1.519868415,.1),
circle(1.5,0,.1), circle(-1.5,0,.1), locate(1.4,1.1,d[2]),
green(line(1.2,1.519868415,-1.5,0),line(1.2,1.519868415,1.5,0)),
locate(-.6,1.1,d[1]), red(circle(2.5,0,2))

 )}}}

Edwin</pre>