The correct option is a = 2b.

Step-by-step Explanation:

In linear programming, if an objective function
$Z=ax+by$
attains its maximum value at more than one corner point, it also attains that same maximum value at every point on the line segment connecting those corner points.

Since we are given that the objective function has its maximum value on the line segment
$AB$
connecting the corner points
$A(0,8)$
and
$B(4,0)$,
the value of the objective function must be equal at both points $A$ and $B$.

Therefore, we can write:
$Z\left(A\right)=Z\left(B\right)$

Let us calculate the value of $Z$ at point
$A(0,8)$:
$Z\left(A\right)=a\left(0\right)+b\left(8\right)=8b$

Let us calculate the value of $Z$ at point
$B(4,0)$:
$Z\left(B\right)=a\left(4\right)+b\left(0\right)=4a$

Now, equating the two values to establish the relation:
$8b=4a$

Dividing both sides of the equation by 4, we get:
$2b=a$
Or, written equivalently:
$a=2b$