Introduction:
This question is based on the concept of sub duplicate ratio. We have to find x^2 when the sub duplicate ratio of p:q is given.

Explanation:
Sub duplicate ratio is a type of ratio in which the square root of the first term is subtracted from the square root of the second term. This concept is used in geometry, physics, and mathematics.

Let's assume that the sub duplicate ratio of p:q is k. Then, we can write:

k = sqrt(q) - sqrt(p)

Now, we have to find the value of x^2 when the sub duplicate ratio of p-x^2:q-x^2 is k.

Solution:

Step 1: Find the sub duplicate ratio of p-x^2:q-x^2.

Sub duplicate ratio of p-x^2:q-x^2 is:

k = sqrt(q-x^2) - sqrt(p-x^2)

Step 2: Substitute the value of k.

We know that:

k = sqrt(q) - sqrt(p)

Substituting the value of k, we get:

sqrt(q-x^2) - sqrt(p-x^2) = sqrt(q) - sqrt(p)

Step 3: Simplify the equation.

Squaring both sides of the equation, we get:

q-x^2 + p-x^2 - 2sqrt((q-x^2)(p-x^2)) = q + p - 2sqrt(pq)

Simplifying the equation, we get:

-2x^2 + 2sqrt((q-x^2)(p-x^2)) = 2sqrt(pq)

Squaring again, we get:

4x^4 - 4x^2(p+q) + 4pq = 4(q-x^2)(p-x^2)

Simplifying, we get:

4x^4 - 4px^2 - 4qx^2 + 4pq = 4px^2 - 4x^4 + 4qx^2 - 4x^4

8x^4 - 8px^2 - 8qx^2 + 8pq = 0

Dividing by 8, we get:

x^4 - px^2 - qx^2 + pq = 0

Step 4: Find x^2.

We know that the sub duplicate ratio of p:q is k. Substituting the value of k, we get:

k = sqrt(q) - sqrt(p)

Squaring both sides, we get:

k^2 = q + p - 2sqrt(pq)

Simplifying, we get:

2sqrt(pq) = p + q - k^2

Squaring again, we get:

4pq = p^2 + q^2 + k^4 - 2pk^2 - 2qk^2 + 2pq

Simplifying, we get:

2pq - p^2 - q^2 + k^4 - 2pk^2 - 2qk^2 = 0

Substituting the value of k, we get:

2pq - p^2 - q^2 + (q-p)^2 = 0

Simplifying, we get:

2pq - p^2 - q^