Given three points (5,2,4),(6,-1,2) and (8,-7,k) and we need to find the value of k if the points are collinear.

Collinear points
Collinear points are the points that lie on the same straight line. In other words, if three or more points are collinear, then the distance between any two points on the line will be same and can be expressed as a linear equation.

Finding the value of k
Let's assume that the three given points are collinear. We can use the distance formula to find the distance between the first two points and the first and third points. If the distances are equal, then the points are collinear.

Distance between (5,2,4) and (6,-1,2) is given by
d1 = sqrt((6-5)^2 + (-1-2)^2 + (2-4)^2) = sqrt(1 + 9 + 4) = sqrt(14)

Distance between (5,2,4) and (8,-7,k) is given by
d2 = sqrt((8-5)^2 + (-7-2)^2 + (k-4)^2) = sqrt(9 + 81 + (k-4)^2)

Now, we need to find the value of k such that d1 = d2

Squaring both sides of the equation, we get
14 = 9 + 81 + (k-4)^2
Simplifying the equation, we get
(k-4)^2 = -76
Taking the square root of both sides, we get
k-4 = sqrt(-76)
k-4 = i*sqrt(76)
k-4 = i*2*sqrt(19)
k = 4 + i*2*sqrt(19)

Since k is not a real number, the given points are not collinear. Therefore, the correct answer is option A (-2) which is not equal to the calculated value of k.