In each of the following questions two equations numbered I and II are given. You have to solve both the equations and mark the appropriate option. Give answer
I `x^2-18x-63=0`
II `y^2-8y-48=0`

Options

A. if `xlty`
B. if `xgey`
C. if `xley`
D. if x=y or relationship between x and y cannot be determined.

Answer

To solve the given equations, we will follow these steps:

### Step 1: Solve the first equation \( x^2 - 18x - 63 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -18 \), and \( c = -63 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \( ac = 1 \times -63 = -63 \) and add up to \( b = -18 \). The numbers are \( -21 \) and \( 3 \).
3. **Rewrite the equation**:
\[ x^2 - 21x + 3x - 63 = 0 \]
4. **Group the terms**:
\[ (x^2 - 21x) + (3x - 63) = 0 \]
5. **Factor by grouping**:
\[ x(x - 21) + 3(x - 21) = 0 \]
\[ (x - 21)(x + 3) = 0 \]
6. **Find the values of \( x \)**:
\[ x - 21 = 0 \quad \Rightarrow \quad x = 21 \]
\[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

### Step 2: Solve the second equation \( y^2 - 8y - 48 = 0 \)

1. **Identify the coefficients**: Here, \( a = 1 \), \( b = -8 \), and \( c = -48 \).
2. **Factor the quadratic equation**: We need to find two numbers that multiply to \( ac = 1 \times -48 = -48 \) and add up to \( b = -8 \). The numbers are \( -12 \) and \( 4 \).
3. **Rewrite the equation**:
\[ y^2 - 12y + 4y - 48 = 0 \]
4. **Group the terms**:
\[ (y^2 - 12y) + (4y - 48) = 0 \]
5. **Factor by grouping**:
\[ y(y - 12) + 4(y - 12) = 0 \]
\[ (y - 12)(y + 4) = 0 \]
6. **Find the values of \( y \)**:
\[ y - 12 = 0 \quad \Rightarrow \quad y = 12 \]
\[ y + 4 = 0 \quad \Rightarrow \quad y = -4 \]

### Step 3: Compare the values of \( x \) and \( y \)

1. **Values of \( x \)**: \( 21, -3 \)
2. **Values of \( y \)**: \( 12, -4 \)

- **First comparison**: \( x = -3 \) and \( y = -4 \)
- Here, \( -3 > -4 \) (since in negative numbers, the number closer to zero is greater).

- **Second comparison**: \( x = 21 \) and \( y = 12 \)
- Here, \( 21 > 12 \).

### Conclusion

In both cases, we see that:
- When \( x = -3 \) and \( y = -4 \), \( x > y \).
- When \( x = 21 \) and \( y = 12 \), \( x > y \).

Thus, the relationship between \( x \) and \( y \) cannot be determined as it varies based on the values chosen.

### Final Answer
The relationship between \( x \) and \( y \) cannot be determined.

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