The equation (x-2)y+(y-5)²=169 describes the circular pattern of a sprinkler set in the middle of a lawn.
To see why, we can rewrite the equation in standard form for a circle:
(x-2)y+(y-5)²=169
xy - 2y + y² - 10y + 25 = 169
x(y-2) + y² - 10y - 144 = 0
(x-(-2))(y-5)² = 144
(x+2)(y-5)² = 144
This is the equation of a circle with center (-2, 5) and radius 12. Therefore, the area of the lawn that gets sprayed by the sprinkler is a circle with center (-2, 5) and radius 12.