Question

One more math problem! I promise this is the last one!!!


A gardener wants three rosebushes in her garden to be watered by a rotating water sprinkler. The gardener draws a diagram of the garden using a grid in which each unit represents 1 ft. The rosebushes are at (1, 3), (5, 11), and (11, 4). She wants to position the sprinkler at a point equidistant from each rosebush. Where should the gardener place the sprinkler? What equation describes the boundary of the circular region that the sprinkler will cover?

Answer 1

Let the position of the sprinkler = (x, y)
=> distance between (x, y) and the given three points will be all equal
=> (x - 1)^2 + (y - 3)^2 = (x - 5)^2 + (y - 11)^2
=> 8x + 16y = 136
=> x + 2y = 17 ... ( 1 )
Also,
(x - 1)^2 + (y - 3)^2 = (x - 11)^2 + (y - 4)^2
=> 20x + 2y = 127 ... ( 2 )
Subtracting eqn. ( 1 ) from ( 2 )
=> 19x = 110
=> x = 110/19
Plugging x = 110/19 in eqn. ( 1 ),
110/19 + 2y = 17
=> 2y = 17 - 110/19
=> y = 213/38
=> Sprinkler should be placed at (110/19, 213/38).