Consider point A(6, 30), point B(24, 6) and line AB: 4x+3y = 114.
Point `P(0, lambda)` is a point on y-axis such that `0 lt lambda lt 38 " and point " Q(0, lambda)` is a point on y-axis such that `lambda gt 38`.
The maximum value of angle APB is

Options

A. `(pi)/(3)`
B. `(pi)/(2)`
C. `(2pi)/(3)`
D. `(3pi)/(3)`

Explanation

To solve the problem, we need to find the maximum value of the angle \( \angle APB \) where points \( A(6, 30) \), \( B(24, 6) \), and \( P(0, \lambda) \) are given, with the conditions on \( \lambda \).

### Step 1: Find the slopes of lines AP and BP

1. **Calculate the slope of line AP**:
- The coordinates of point A are \( (6, 30) \) and point P is \( (0, \lambda) \).
- The slope \( m_1 \) of line AP is given by:
\[ m_1 = \frac{30 - \lambda}{6 - 0} = \frac{30 - \lambda}{6} \]

2. **Calculate the slope of line BP**:
- The coordinates of point B are \( (24, 6) \) and point P is \( (0, \lambda) \).
- The slope \( m_2 \) of line BP is given by:
\[ m_2 = \frac{6 - \lambda}{24 - 0} = \frac{6 - \lambda}{24} \]

### Step 2: Use the formula for the tangent of the angle between two lines

The tangent of the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is given by:
\[ \tan \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \]

Substituting the values of \( m_1 \) and \( m_2 \):
\[ \tan \theta = \frac{\frac{30 - \lambda}{6} - \frac{6 - \lambda}{24}}{1 + \left(\frac{30 - \lambda}{6}\right) \left(\frac{6 - \lambda}{24}\right)} \]

### Step 3: Simplify the expression

1. **Find a common denominator for the numerator**:
\[ \tan \theta = \frac{\frac{(30 - \lambda) \cdot 4 - (6 - \lambda)}{24}}{1 + \frac{(30 - \lambda)(6 - \lambda)}{144}} \]

Simplifying the numerator:
\[ = \frac{(120 - 4\lambda - 6 + \lambda)}{24} = \frac{114 - 3\lambda}{24} \]

2. **Denominator**:
\[ = 1 + \frac{(30 - \lambda)(6 - \lambda)}{144} \]

### Step 4: Set conditions for maximum angle

To maximize \( \tan \theta \), we need to consider the conditions on \( \lambda \):
- For \( 0 < \lambda < 38 \), we want to find the maximum value of \( \tan \theta \).

### Step 5: Analyze the behavior of \( \tan \theta \)

As \( \lambda \) approaches 18, we can see that the value of \( \tan \theta \) tends to infinity. This indicates that the angle \( \theta \) approaches \( \frac{\pi}{2} \).

### Conclusion

Thus, the maximum value of angle \( \angle APB \) is:
\[ \theta = \frac{\pi}{2} \]