Understanding the Problem
To find the ratio in which the line segment joining the points \( A(2, 3, 4) \) and \( B(-1, 4, 5) \) is divided by the plane \( 3x + 2y - z - 2 = 0 \), we need to determine where the line intersects the plane and then use section formula to find the ratio.
Step 1: Equation of the Line
The parametric equations of the line joining points \( A \) and \( B \) can be expressed as:
- \( x = 2 + t(-1 - 2) = 2 - 3t \)
- \( y = 3 + t(4 - 3) = 3 + t \)
- \( z = 4 + t(5 - 4) = 4 + t \)
where \( t \) is a parameter.
Step 2: Substitute into the Plane Equation
Now, substitute the parametric equations into the plane equation:
\[ 3(2 - 3t) + 2(3 + t) - (4 + t) - 2 = 0 \]
This simplifies to:
\[ 6 - 9t + 6 + 2t - 4 - t - 2 = 0 \]
Combine like terms:
\[ -8t + 6 = 0 \]
Solving for \( t \):
\[ t = \frac{3}{4} \]
Step 3: Finding the Point of Intersection
Substituting \( t = \frac{3}{4} \) back into the line equations:
- \( x = 2 - 3 \times \frac{3}{4} = 2 - \frac{9}{4} = -\frac{1}{4} \)
- \( y = 3 + \frac{3}{4} = \frac{15}{4} \)
- \( z = 4 + \frac{3}{4} = \frac{19}{4} \)
The point of intersection is \( P\left(-\frac{1}{4}, \frac{15}{4}, \frac{19}{4}\right) \).
Step 4: Ratio Calculation
Using the section formula, the ratio \( k:1 \) can be calculated:
- Let \( A(2, 3, 4) \) correspond to \( k \) and \( B(-1, 4, 5) \) correspond to \( 1 \).
- The coordinates of point \( P \) give:
\[
\frac{(-1/4 - 2)}{(-1 - 2)} = \frac{-9/4}{-3} = \frac{3}{4}
\]
Thus, the ratio in which the line segment is divided is \( 3:1 \).