If x,y, z and w are non-zero real numbers and
` x^(2) + 5y^(2) + 5z^2+4w^(2) - 4xy - 4yz - 4zw = 0 ` , then
x, y, z,w are in

Options

A. A.P.
B. A.G.P
C. H.P.
D. G.P.

Answer

To solve the problem, we start with the equation given:

\[ x^2 + 5y^2 + 5z^2 + 4w^2 - 4xy - 4yz - 4zw = 0 \]

### Step 1: Rewrite the equation
We can rewrite \( 5y^2 \) and \( 5z^2 \) as follows:

\[ 5y^2 = 4y^2 + y^2 \]
\[ 5z^2 = 4z^2 + z^2 \]

Substituting these back into the equation gives:

\[ x^2 + (4y^2 + y^2) + (4z^2 + z^2) + 4w^2 - 4xy - 4yz - 4zw = 0 \]

### Step 2: Group the terms
Now, we can group the terms in pairs:

\[ x^2 - 4xy + 4y^2 + y^2 + 4z^2 - 4yz + z^2 + 4w^2 - 4zw = 0 \]

This can be rearranged as:

\[ (x - 2y)^2 + (y - 2z)^2 + (z - 2w)^2 = 0 \]

### Step 3: Analyze the equation
Since the sum of squares is equal to zero, each individual square must also be zero:

1. \( (x - 2y)^2 = 0 \)
2. \( (y - 2z)^2 = 0 \)
3. \( (z - 2w)^2 = 0 \)

### Step 4: Solve the equations
From these equations, we can deduce:

1. \( x - 2y = 0 \) → \( x = 2y \)
2. \( y - 2z = 0 \) → \( y = 2z \)
3. \( z - 2w = 0 \) → \( z = 2w \)

### Step 5: Express all variables in terms of \( w \)
Now we can express \( x, y, z \) in terms of \( w \):

- From \( z = 2w \), we substitute into \( y = 2z \):
\[ y = 2(2w) = 4w \]

- Then substitute \( y = 4w \) into \( x = 2y \):
\[ x = 2(4w) = 8w \]

### Step 6: Write down the values
Now we have:

- \( x = 8w \)
- \( y = 4w \)
- \( z = 2w \)
- \( w = w \)

### Step 7: Identify the relationship
Now we can see that:

- \( x, y, z, w \) can be expressed as:
\[ x = 8w, \quad y = 4w, \quad z = 2w, \quad w = w \]

This forms a geometric progression (GP) where:

- The first term \( a = 8w \)
- The common ratio \( r = \frac{y}{x} = \frac{4w}{8w} = \frac{1}{2} \)

### Conclusion
Thus, \( x, y, z, w \) are in a geometric progression (GP).