if m and b are real numbers and `mbgt0`, then the line whose equation is `y=mx+b` cannot contain the point

Options

A. ` ( 0, 2008)`
B. `(2008, 0)`
C. `(0, -2008)`
D. `(20, -100)`

Answer

To solve the problem, we need to determine which point cannot lie on the line represented by the equation \( y = mx + b \) given that \( mb > 0 \). This condition implies two scenarios: either both \( m \) and \( b \) are positive, or both \( m \) and \( b \) are negative.

### Step-by-Step Solution:

1. **Understanding the Condition \( mb > 0 \)**:
- If \( m > 0 \) and \( b > 0 \): The line has a positive slope and a positive y-intercept.
- If \( m < 0 \) and \( b < 0 \): The line has a negative slope and a negative y-intercept.

2. **Sketching the Lines**:
- For \( m > 0 \) and \( b > 0 \):
- The line will rise from the y-intercept \( b \) (above the origin) and will continue to rise as \( x \) increases.
- For \( m < 0 \) and \( b < 0 \):
- The line will fall from the y-intercept \( b \) (below the origin) and will continue to fall as \( x \) increases.

3. **Identifying Points**:
- We need to analyze the given points to check if they can lie on either of the lines.
- The points to consider are:
1. \( (0, 2008) \)
2. \( (a, 0) \) for some positive \( a \)
3. \( (0, -2008) \)
4. \( (20, -100) \)

4. **Analyzing Each Point**:
- **Point 1**: \( (0, 2008) \)
- This point can lie on the line if \( b = 2008 \) (which is possible if \( m > 0 \)).
- **Point 2**: \( (a, 0) \) where \( a > 0 \)
- This point can lie on the line if \( y = 0 \) when \( x = a \). This is possible for both cases.
- **Point 3**: \( (0, -2008) \)
- This point can lie on the line if \( b = -2008 \) (which is possible if \( m < 0 \)).
- **Point 4**: \( (20, -100) \)
- This point has a positive x-coordinate and a negative y-coordinate, which means it can only lie on the line if \( m < 0 \) and \( b < 0 \). However, since \( m \) and \( b \) cannot be both negative in the first case, this point cannot lie on the line when \( mb > 0 \).

5. **Conclusion**:
- The point that cannot be contained by the line \( y = mx + b \) under the condition \( mb > 0 \) is \( (a, 0) \) for some positive \( a \).

### Final Answer:
The line whose equation is \( y = mx + b \) cannot contain the point \( (a, 0) \) for \( a > 0 \).