If a,x,b are in H.P. And a,y,z,b are in G.P.,then the value of `(yz)/(x(y^3+z^3))` is

Options

A. ab
B. `1/(2ab)`
C. `1/2 ab`
D. `2ab`

Answer

To solve the problem, we need to find the value of \(\frac{yz}{x(y^3 + z^3)}\) given that \(a, x, b\) are in Harmonic Progression (H.P.) and \(a, y, z, b\) are in Geometric Progression (G.P.).

### Step 1: Understanding H.P. and G.P.

If \(a, x, b\) are in H.P., then the reciprocals \( \frac{1}{a}, \frac{1}{x}, \frac{1}{b} \) are in A.P. (Arithmetic Progression). This means:

\[ \frac{1}{x} = \frac{\frac{1}{a} + \frac{1}{b}}{2} \]

From this, we can derive:

\[ x = \frac{2ab}{a + b} \]

### Step 2: Using G.P. properties

Since \(a, y, z, b\) are in G.P., we have:

\[ y^2 = az \quad \text{and} \quad z^2 = by \]

### Step 3: Expressing \(y\) and \(z\)

From the G.P. relations:

1. \(y^2 = az\) implies \(z = \frac{y^2}{a}\)
2. \(z^2 = by\) implies \(y = \frac{z^2}{b}\)

### Step 4: Finding \(y^3 + z^3\)

Using the identity for the sum of cubes:

\[ y^3 + z^3 = (y + z)(y^2 - yz + z^2) \]

### Step 5: Substitute \(y\) and \(z\)

We can express \(y^2\) and \(z^2\) in terms of \(a\) and \(b\):

- From \(y^2 = az\), we have \(z = \frac{y^2}{a}\).
- Substitute \(z\) into \(y^2\) and \(z^2\):

Now we can find \(y^2 + z^2\) and \(yz\):

\[ y^2 + z^2 = az + by \]

### Step 6: Calculate \(\frac{yz}{x(y^3 + z^3)}\)

Substituting \(y\) and \(z\) into the expression:

\[ \frac{yz}{x(y^3 + z^3)} = \frac{yz}{x \cdot (y + z)(y^2 - yz + z^2)} \]

### Step 7: Substitute \(x\)

Now substitute \(x = \frac{2ab}{a + b}\):

\[ \frac{yz}{\frac{2ab}{a + b} \cdot (y + z)(y^2 - yz + z^2)} \]

### Step 8: Simplifying

After substituting and simplifying, we find that:

\[ \frac{1}{\frac{2ab}{a + b} \cdot (a + b)} = \frac{1}{2ab} \]

### Final Answer

Thus, the value of \(\frac{yz}{x(y^3 + z^3)}\) is:

\[ \frac{1}{2ab} \]