So point C is sliding only along the line AB.
The warning then will sound if point C is at 0.5 units to the right of point A.
Let's call point C' the position of point C when it is 0.5 units to the right of A.
And point D' the position of point D at that instant.
If we can get the x-coordinate of C', then that is also the x-coordinate of D'. Then we can get the y-coordinate of D' by using the circle equation.
Draw a radius to A.
By Pythagorean theorem,
4^2 = (3 -2)^2 +(AC)^2
AC = sqrt(16 -1) = 3.87298
So the coordinates of A are ((1 -3.87298),3) = (-2.87298,3)
Then the position of C' is ((-2.87298 +0.5),3) = (-2.37298,3)
Hence the x-coordinate of D' is -2.37298 also.
In the circle,
(x -1)^2 +(y -2)^2 = 16,
(-2.37298 -1)^2 +(y -2)^2 = 16
(y -2)^2 = 16 -(-3.37298)^2
y -2 = sqrt(16 -11.37699)
y = 2.15012 +2
y = 4.15012
Hence, D' is (-2.37298,4.15012)
Therefore, whenever D is between, and including, points (-2,37298,4.15012) and (-2.87298,3) along the circumference of the circle, the alarm rings. ----answer.