To solve this problem, we need to use the given information that a, b, c are in arithmetic progression (AP) and x, y, z are in geometric progression (GP). We are also given the expression xb - c. yc - a. za - b and we need to determine its value.

Let's start by writing the given arithmetic and geometric progressions:

Arithmetic Progression:
a, b, c

Geometric Progression:
x, y, z

Now, let's find the common difference (d) for the arithmetic progression and the common ratio (r) for the geometric progression:

Common Difference (d):
d = b - a
d = c - b

Common Ratio (r):
r = y / x
r = z / y

To simplify the expression xb - c. yc - a. za - b, we can substitute the values of b and c in terms of a and d, and the values of y and z in terms of x and r:

xb - c. yc - a. za - b
= (a + d)b - c. (x + rx)y - a. (x + rx)(x + rx) - b

Now, let's expand and simplify this expression:

(a + d)b - c. (x + rx)y - a. (x + rx)(x + rx) - b
= ab + db - cy - cry + axy + adxy - ax^2 - ar^2x^2 - 2arx^3 - b

We can further simplify this expression by grouping the terms:

ab + db - cy - cry + axy + adxy - ax^2 - ar^2x^2 - 2arx^3 - b
= (ab - b) + (db - cy) + (axy - ax^2) + (adxy - ar^2x^2 - 2arx^3)

Now, let's factor out the common terms from each group:

(ab - b) + (db - cy) + (axy - ax^2) + (adxy - ar^2x^2 - 2arx^3)
= b(a - 1) + d(b - c) + ax(y - x) + adxy(1 - r^2 - 2rx)

Since a, b, c are in arithmetic progression and x, y, z are in geometric progression, we know that a - 1 = 0 and 1 - r^2 - 2rx = 0. Therefore, the expression simplifies to:

b(a - 1) + d(b - c) + ax(y - x) + adxy(1 - r^2 - 2rx)
= b(0) + d(b - c) + ax(y - x) + a(0)(0)
= 0 + d(b - c) + ax(y - x)
= d(b - c) + ax(y - x)

Since d = b - a and y - x = r(x - y), we can substitute these values into the expression:

d(b - c) + ax(y - x)
= (b - a)(b - c) + ax(r(x - y))
= (b - a)(b - c) - axr(y - x)

Now, let's simplify this expression further