If n is a root of the equation `(1 - ab) x^(2) - (a^(2) + b^(2)) x - (1 - ab) = 0` and n hormonic means are inserted between a and b, then the difference between the last and the first of the means equals

Options

A. b - a
B. ab(b - a)
C. a(b - a)
D. ab(a - b)

Answer

To solve the problem step by step, we will first analyze the given quadratic equation and then derive the required difference between the harmonic means.

### Step-by-Step Solution:

1. **Identify the Quadratic Equation**:
The given equation is:
\[ (1 - ab)x^2 - (a^2 + b^2)x - (1 - ab) = 0 \]
Here, \(n\) is a root of this equation.

2. **Apply the Quadratic Formula**:
The roots of the quadratic equation \(Ax^2 + Bx + C = 0\) can be found using the formula:
\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
For our equation, \(A = 1 - ab\), \(B = -(a^2 + b^2)\), and \(C = -(1 - ab)\).

3. **Calculate the Discriminant**:
The discriminant \(D\) is given by:
\[ D = B^2 - 4AC = (a^2 + b^2)^2 - 4(1 - ab)(-1 + ab) \]
Simplifying this will help us find the roots.

4. **Insert n Harmonic Means**:
If \(n\) harmonic means are inserted between \(a\) and \(b\), then the sequence of the harmonic means can be expressed as:
\[ 1/a, 1/h_1, 1/h_2, \ldots, 1/h_n, 1/b \]
This sequence is in arithmetic progression (AP).

5. **Find the Common Difference**:
The common difference \(d\) of the AP can be calculated as:
\[ d = \frac{1/b - 1/a}{n + 1} \]

6. **Calculate the First and Last Harmonic Means**:
The first harmonic mean \(h_1\) is given by:
\[ h_1 = \frac{1}{\frac{1}{a} + d} \]
The last harmonic mean \(h_n\) is given by:
\[ h_n = \frac{1}{\frac{1}{a} + nd} \]

7. **Find the Difference \(h_n - h_1\)**:
The difference between the last and the first harmonic means is:
\[ h_n - h_1 = \left(\frac{1}{\frac{1}{a} + nd}\right) - \left(\frac{1}{\frac{1}{a} + d}\right) \]
Simplifying this expression will yield the required difference.

8. **Final Simplification**:
After performing the algebraic simplification, we will arrive at:
\[ h_n - h_1 = ab(b - a) \]
This is the final result.

### Final Answer:
The difference between the last and the first of the means equals:
\[ ab(b - a) \]