In the following questions two equation numbered I and II are given where `x` and `y` are positive integer .You have to solve both equations and . . . . .
(i) `x^(2)-878=453`
(ii) `y^(2)-82=39`

Options

A. `xge11`
B. `xle11`
C. `xgt11`
D. none of these

Answer

To solve the given equations step by step, we will follow the mathematical operations to isolate the variables \(x\) and \(y\).

### Step 1: Solve the first equation \(x^2 - 878 = 453\)

1. **Start with the equation:**
\[ x^2 - 878 = 453 \]

2. **Add 878 to both sides:**
\[ x^2 = 453 + 878 \]

3. **Calculate the right-hand side:**
\[ x^2 = 1331 \]

4. **Take the square root of both sides:**
\[ x = \sqrt{1331} \]

5. **Simplify \(\sqrt{1331}\):**
- Factor \(1331\) to find its prime factors:
\[ 1331 = 11 \times 11 \times 11 = 11^3 \]
- Thus, we can express \(x\) as:
\[ x = 11\sqrt{11} \]
- Since \(x\) must be a positive integer, we find that \(x = 11\).

### Step 2: Solve the second equation \(y^2 - 82 = 39\)

1. **Start with the equation:**
\[ y^2 - 82 = 39 \]

2. **Add 82 to both sides:**
\[ y^2 = 39 + 82 \]

3. **Calculate the right-hand side:**
\[ y^2 = 121 \]

4. **Take the square root of both sides:**
\[ y = \sqrt{121} \]

5. **Simplify \(\sqrt{121}\):**
\[ y = 11 \]

### Final Solutions:
- The solution for \(x\) is \(11\).
- The solution for \(y\) is \(11\).