The point P(a, b) is such that `b-25a=4` and the arithmetic mean of a and b is 28. Q(x, y) is the point such that x and y are two geometric means between a and b, if O is the origin then `(OP)^(2)+(OQ)^(2)` is equal to

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) based on the given conditions and then calculate \( (OP)^2 + (OQ)^2 \).

### Step 1: Set up the equations
We are given two equations:
1. \( b - 25a = 4 \) (Equation 1)
2. The arithmetic mean of \( a \) and \( b \) is 28, which gives us:
\[ \frac{a + b}{2} = 28 \implies a + b = 56 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations
From Equation 2, we can express \( b \) in terms of \( a \):
\[ b = 56 - a \]

Now, substitute this expression for \( b \) into Equation 1:
\[ (56 - a) - 25a = 4 \]
Simplifying this gives:
\[ 56 - 26a = 4 \]
\[ -26a = 4 - 56 \]
\[ -26a = -52 \implies a = 2 \]

### Step 3: Find \( b \)
Now that we have \( a \), we can find \( b \):
\[ b = 56 - a = 56 - 2 = 54 \]

### Step 4: Identify point \( P \)
Thus, the coordinates of point \( P \) are:
\[ P(2, 54) \]

### Step 5: Find geometric means \( x \) and \( y \)
Since \( x \) and \( y \) are two geometric means between \( a \) and \( b \), we can express them in terms of a common ratio \( r \):
- The first term is \( a = 2 \)
- The second term is \( x = 2r \)
- The third term is \( y = 2r^2 \)
- The fourth term is \( b = 54 \)

From the last term, we have:
\[ 54 = 2r^3 \implies r^3 = 27 \implies r = 3 \]

### Step 6: Calculate \( x \) and \( y \)
Now substituting \( r \) back to find \( x \) and \( y \):
\[ x = 2r = 2 \times 3 = 6 \]
\[ y = 2r^2 = 2 \times 3^2 = 2 \times 9 = 18 \]

### Step 7: Identify point \( Q \)
Thus, the coordinates of point \( Q \) are:
\[ Q(6, 18) \]

### Step 8: Calculate \( (OP)^2 \) and \( (OQ)^2 \)
Now we will calculate \( (OP)^2 \) and \( (OQ)^2 \):
- For point \( P(2, 54) \):
\[ (OP)^2 = (2 - 0)^2 + (54 - 0)^2 = 2^2 + 54^2 = 4 + 2916 = 2920 \]

- For point \( Q(6, 18) \):
\[ (OQ)^2 = (6 - 0)^2 + (18 - 0)^2 = 6^2 + 18^2 = 36 + 324 = 360 \]

### Step 9: Find \( (OP)^2 + (OQ)^2 \)
Finally, we add the two results:
\[ (OP)^2 + (OQ)^2 = 2920 + 360 = 3280 \]

### Final Answer
Thus, the value of \( (OP)^2 + (OQ)^2 \) is:
\[ \boxed{3280} \]