To solve the problem of finding the ratio in which the XZ plane divides the line segment joining the points \( P(3, 2, b) \) and \( A(a, -4, 3) \), we can follow these steps:
### Step 1: Understand the Concept of Division of a Line Segment
The XZ plane is defined by the equation \( y = 0 \). This means that when the line segment is divided by the XZ plane, the y-coordinate of the point of division must be zero.
### Step 2: Use the Section Formula
Let the point \( P \) divide the line segment \( AB \) in the ratio \( k:1 \). The coordinates of point \( P \) can be calculated using the section formula:
\[
P(x, y, z) = \left( \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}, \frac{k \cdot z_2 + 1 \cdot z_1}{k + 1} \right)
\]
Where \( A(x_1, y_1, z_1) = (3, 2, b) \) and \( B(x_2, y_2, z_2) = (a, -4, 3) \).
### Step 3: Calculate the y-coordinate of Point P
Using the coordinates of points \( A \) and \( B \):
\[
y = \frac{k \cdot (-4) + 1 \cdot 2}{k + 1}
\]
Setting \( y = 0 \) for the XZ plane:
\[
0 = \frac{k \cdot (-4) + 2}{k + 1}
\]
### Step 4: Solve for k
To eliminate the fraction, we can multiply both sides by \( k + 1 \):
\[
0 = k \cdot (-4) + 2
\]
This simplifies to:
\[
4k = 2 \quad \Rightarrow \quad k = \frac{2}{4} = \frac{1}{2}
\]
### Step 5: Determine the Ratio
The ratio in which the XZ plane divides the line segment is \( k:1 \), which translates to:
\[
\frac{1}{2}:1 \quad \Rightarrow \quad 1:2
\]
### Conclusion
Thus, the XZ plane divides the line segment joining the points \( (3, 2, b) \) and \( (a, -4, 3) \) in the ratio \( 1:2 \).
