Statement 1 : Let `f : r-{3}` be a function given by , `f(x+10)=(f(x)-5)/(f(x)-3)," then " f(10)=f(50)`.
Statement 2 : f (x) is a periodic function.

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we need to analyze the function given by the equation \( f(x + 10) = \frac{f(x) - 5}{f(x) - 3} \) and determine if \( f(10) = f(50) \) holds true, as well as whether \( f(x) \) is a periodic function.

### Step-by-Step Solution:

1. **Substituting \( x \) with \( x + 10 \)**:
\[ f(x + 20) = \frac{f(x + 10) - 5}{f(x + 10) - 3} \]
Using the original equation, replace \( f(x + 10) \):
\[ f(x + 20) = \frac{\frac{f(x) - 5}{f(x) - 3} - 5}{\frac{f(x) - 5}{f(x) - 3} - 3} \]

2. **Simplifying the Right Side**:
To simplify, we need to find a common denominator:
\[ f(x + 20) = \frac{\frac{f(x) - 5 - 5(f(x) - 3)}{f(x) - 3}}{\frac{f(x) - 5 - 3(f(x) - 3)}{f(x) - 3}} \]
Simplifying the numerator:
\[ f(x + 20) = \frac{f(x) - 5 - 5f(x) + 15}{f(x) - 5 - 3f(x) + 9} \]
This simplifies to:
\[ f(x + 20) = \frac{-4f(x) + 10}{-2f(x) + 4} \]

3. **Further Simplifying**:
Dividing both the numerator and denominator by 2:
\[ f(x + 20) = \frac{-2f(x) + 5}{-f(x) + 2} \]

4. **Substituting \( x \) with \( x + 20 \)**:
Now, substitute \( x \) with \( x + 20 \):
\[ f(x + 40) = \frac{-2f(x + 20) + 5}{-f(x + 20) + 2} \]
Using the previous result for \( f(x + 20) \):
\[ f(x + 40) = \frac{-2\left(\frac{-2f(x) + 5}{-f(x) + 2}\right) + 5}{- \left(\frac{-2f(x) + 5}{-f(x) + 2}\right) + 2} \]

5. **Simplifying Again**:
This will yield:
\[ f(x + 40) = \frac{4f(x) - 10 + 5(-f(x) + 2)}{2 + 2f(x) - 5} \]
After simplification, we find:
\[ f(x + 40) = f(x) \]

6. **Conclusion**:
Since \( f(x + 40) = f(x) \), we conclude that \( f(x) \) is periodic with a period of 40. Therefore, \( f(10) = f(50) \) holds true.

### Final Statements:
- **Statement 1**: \( f(10) = f(50) \) is true.
- **Statement 2**: \( f(x) \) is a periodic function is also true.