Assertion (A): The locus of the point `((e^(2t)+e^(-2t))/(2), (e^(2t)-e^(-2t))/(2))` when 't' is a parameter represents a rectangular hyperbola.
Reason (R ) : The eccentricity of a rectangular hyperbola is 2.

Options

A. a) Both A and R are true and R is the correct explanation of A
B. b) Both A and R are true and R is not the correct explanation of A
C. c) A is true, R is false
D. d) A is false, R is true

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question.

### Step 1: Identify the point representation
The point given is:
\[ P\left(x, y\right) = \left(\frac{e^{2t} + e^{-2t}}{2}, \frac{e^{2t} - e^{-2t}}{2}\right) \]
We will denote:
\[ x = \frac{e^{2t} + e^{-2t}}{2} \]
\[ y = \frac{e^{2t} - e^{-2t}}{2} \]

### Step 2: Express \(e^{2t}\) and \(e^{-2t}\) in terms of \(x\) and \(y\)
From the expressions for \(x\) and \(y\), we can express \(e^{2t}\) and \(e^{-2t}\):
1. Adding the equations:
\[ x + y = \frac{e^{2t} + e^{-2t}}{2} + \frac{e^{2t} - e^{-2t}}{2} = e^{2t} \]
Thus, \(e^{2t} = x + y\).

2. Subtracting the equations:
\[ x - y = \frac{e^{2t} + e^{-2t}}{2} - \frac{e^{2t} - e^{-2t}}{2} = e^{-2t} \]
Thus, \(e^{-2t} = x - y\).

### Step 3: Formulate the equation
Now, we can express:
\[ e^{2t} \cdot e^{-2t} = (x + y)(x - y) \]
Since \(e^{2t} \cdot e^{-2t} = 1\), we have:
\[ (x + y)(x - y) = 1 \]

### Step 4: Rearranging the equation
Expanding the left-hand side gives:
\[ x^2 - y^2 = 1 \]
This is the standard form of a hyperbola.

### Step 5: Identify the type of hyperbola
The equation \(x^2 - y^2 = 1\) represents a rectangular hyperbola, which is characterized by the property that the transverse and conjugate axes are equal in length.

### Step 6: Analyze the eccentricity
The eccentricity \(e\) of a rectangular hyperbola is given by:
\[ e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{1 + 1} = \sqrt{2} \]
Thus, the eccentricity of a rectangular hyperbola is \(\sqrt{2}\), not 2.

### Conclusion
- The assertion (A) is **true**: The locus represents a rectangular hyperbola.
- The reason (R) is **false**: The eccentricity of a rectangular hyperbola is \(\sqrt{2}\), not 2.

Thus, the correct answer is that assertion (A) is true, but reason (R) is false.