If the area covered by `y=(2)/(x)` and `y=(2)/(2x-1)` from `x=1 or x=e` is ln (a) sq. units, then `(2e-1)^(2)a^(2)` is equal to

Options

A. `(e)/(2)`
B. `e^(4)`
C. 1
D. e

Answer

To solve the problem, we need to find the area between the curves \( y = \frac{2}{x} \) and \( y = \frac{2}{2x - 1} \) from \( x = 1 \) to \( x = e \). We will then relate this area to \( \ln(a) \) and find the value of \( (2e - 1)^2 a^2 \).

### Step-by-Step Solution:

1. **Identify the curves and the area to be calculated:**
We have two curves:
- \( y_1 = \frac{2}{x} \)
- \( y_2 = \frac{2}{2x - 1} \)

We need to find the area between these curves from \( x = 1 \) to \( x = e \).

2. **Set up the integral for the area:**
The area \( A \) between the curves from \( x = 1 \) to \( x = e \) can be expressed as:
\[ A = \int_{1}^{e} \left( y_1 - y_2 \right) \, dx = \int_{1}^{e} \left( \frac{2}{x} - \frac{2}{2x - 1} \right) \, dx \]

3. **Simplify the integrand:**
We can factor out the 2:
\[ A = 2 \int_{1}^{e} \left( \frac{1}{x} - \frac{1}{2x - 1} \right) \, dx \]

4. **Integrate each term:**
- The integral of \( \frac{1}{x} \) is \( \ln(x) \).
- For \( \frac{1}{2x - 1} \), we can use substitution. Let \( u = 2x - 1 \), then \( du = 2dx \) or \( dx = \frac{du}{2} \). The limits change as follows:
- When \( x = 1 \), \( u = 1 \)
- When \( x = e \), \( u = 2e - 1 \)
Thus, the integral becomes:
\[ \int \frac{1}{2x - 1} \, dx = \frac{1}{2} \ln(2x - 1) \]

5. **Calculate the definite integrals:**
\[ A = 2 \left[ \ln(x) - \frac{1}{2} \ln(2x - 1) \right]_{1}^{e} \]
Evaluating at the limits:
- At \( x = e \):
\[ \ln(e) - \frac{1}{2} \ln(2e - 1) = 1 - \frac{1}{2} \ln(2e - 1) \]
- At \( x = 1 \):
\[ \ln(1) - \frac{1}{2} \ln(1) = 0 \]
Therefore:
\[ A = 2 \left( 1 - \frac{1}{2} \ln(2e - 1) \right) = 2 - \ln(2e - 1) \]

6. **Relate the area to \( \ln(a) \):**
We are given that \( A = \ln(a) \):
\[ 2 - \ln(2e - 1) = \ln(a) \]
Rearranging gives:
\[ \ln(a) = 2 - \ln(2e - 1) \implies \ln(a) = \ln(e^2) - \ln(2e - 1) \]
Thus:
\[ a = \frac{e^2}{2e - 1} \]

7. **Calculate \( (2e - 1)^2 a^2 \):**
\[ a^2 = \left( \frac{e^2}{2e - 1} \right)^2 = \frac{e^4}{(2e - 1)^2} \]
Therefore:
\[ (2e - 1)^2 a^2 = (2e - 1)^2 \cdot \frac{e^4}{(2e - 1)^2} = e^4 \]

### Final Result:
\[ (2e - 1)^2 a^2 = e^4 \]