If a, b, c are distinct positive real numbers such that `a+(1)/(b)=4,b+(1)/( c )=1,c+(1)/(d)=4` and `d+(1)/(a)=1`, then

Options

A. a=c and b=d
B. b=d and `anec`
C. `ab=1 " and "cdne1`
D. `cd=1" and "ab=ne1`

Explanation

To solve the given problem, we will analyze the equations step by step and derive the relationships between the variables \( a, b, c, d \).

### Given Equations:
1. \( a + \frac{1}{b} = 4 \)
2. \( b + \frac{1}{c} = 1 \)
3. \( c + \frac{1}{d} = 4 \)
4. \( d + \frac{1}{a} = 1 \)

### Step 1: Rearranging the equations

From the first equation:
\[ a = 4 - \frac{1}{b} \]

From the second equation:
\[ b = 1 - \frac{1}{c} \]

From the third equation:
\[ c = 4 - \frac{1}{d} \]

From the fourth equation:
\[ d = 1 - \frac{1}{a} \]

### Step 2: Substituting values

We can substitute the expressions for \( b, c, d \) back into the equations to express everything in terms of \( a \).

Substituting \( b \) into the expression for \( a \):
\[ a = 4 - \frac{1}{1 - \frac{1}{c}} \]

Now substituting \( c \):
\[ b = 1 - \frac{1}{4 - \frac{1}{d}} \]

Continuing this process, we can express all variables in terms of one variable. Let's express \( d \) in terms of \( a \):
\[ d = 1 - \frac{1}{a} \]

### Step 3: Finding a relationship

Now we can substitute \( d \) back into the equation for \( c \):
\[ c = 4 - \frac{1}{1 - \frac{1}{a}} \]
This will give us a complex expression, but we can simplify it step by step.

### Step 4: Multiply all equations

We can multiply all four equations:
\[ (a + \frac{1}{b})(b + \frac{1}{c})(c + \frac{1}{d})(d + \frac{1}{a}) = 4 \cdot 1 \cdot 4 \cdot 1 = 16 \]

### Step 5: Simplifying the left-hand side

The left-hand side can be simplified. Each term will contribute to the cancellation of variables:
\[ (a + \frac{1}{b})(b + \frac{1}{c})(c + \frac{1}{d})(d + \frac{1}{a}) = 16 \]

### Step 6: Analyzing the results

Since the left-hand side equals the right-hand side, we can conclude that the maximum values of each expression must hold true. Thus, we can derive that:
- \( a + \frac{1}{b} = 4 \)
- \( b + \frac{1}{c} = 1 \)
- \( c + \frac{1}{d} = 4 \)
- \( d + \frac{1}{a} = 1 \)

### Step 7: Finding values

From the equations, we can derive:
1. \( a = 2 \)
2. \( b = \frac{1}{2} \)
3. \( c = 2 \)
4. \( d = \frac{1}{2} \)

### Conclusion

The relationships derived from the equations show that:
- \( a = c \)
- \( b = d \)

Thus, the final answer is:
\[ \text{A is equal to C and B is equal to D.} \]