To solve the problem of finding the ratio in which the line segment joining the points \( P(2, 3, 4) \) and \( Q(-3, 5, -4) \) is divided by the \( yz \)-plane, and to find the point of contact, we can follow these steps:
### Step 1: Understand the problem
The \( yz \)-plane is defined by the equation \( x = 0 \). We need to find the point \( R \) on the line segment \( PQ \) such that the \( x \)-coordinate of \( R \) is 0.
### Step 2: Use the section formula
Let the ratio in which the point \( R \) divides the segment \( PQ \) be \( \lambda:1 \). According to the section formula, the coordinates of point \( R \) can be expressed as:
\[
R = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n} \right)
\]
where \( P(x_1, y_1, z_1) = P(2, 3, 4) \) and \( Q(x_2, y_2, z_2) = Q(-3, 5, -4) \).
### Step 3: Substitute the coordinates into the formula
Substituting the coordinates of \( P \) and \( Q \) into the formula, we have:
\[
R = \left( \frac{\lambda \cdot (-3) + 1 \cdot 2}{\lambda + 1}, \frac{\lambda \cdot 5 + 1 \cdot 3}{\lambda + 1}, \frac{\lambda \cdot (-4) + 1 \cdot 4}{\lambda + 1} \right)
\]
### Step 4: Set the \( x \)-coordinate to 0
Since \( R \) lies on the \( yz \)-plane, we set the \( x \)-coordinate to 0:
\[
\frac{-3\lambda + 2}{\lambda + 1} = 0
\]
This implies:
\[
-3\lambda + 2 = 0
\]
### Step 5: Solve for \( \lambda \)
Solving for \( \lambda \):
\[
-3\lambda = -2 \implies \lambda = \frac{2}{3}
\]
### Step 6: Find the ratio
The ratio in which the line segment is divided is \( \lambda:1 = \frac{2}{3}:1 \), which simplifies to:
\[
2:3
\]
### Step 7: Find the coordinates of point \( R \)
Now, substitute \( \lambda = \frac{2}{3} \) back into the section formula to find the coordinates of \( R \):
\[
R_y = \frac{\frac{2}{3} \cdot 5 + 3}{\frac{2}{3} + 1} = \frac{\frac{10}{3} + 3}{\frac{5}{3}} = \frac{\frac{10}{3} + \frac{9}{3}}{\frac{5}{3}} = \frac{\frac{19}{3}}{\frac{5}{3}} = \frac{19}{5}
\]
\[
R_z = \frac{\frac{2}{3} \cdot (-4) + 4}{\frac{2}{3} + 1} = \frac{-\frac{8}{3} + 4}{\frac{5}{3}} = \frac{-\frac{8}{3} + \frac{12}{3}}{\frac{5}{3}} = \frac{\frac{4}{3}}{\frac{5}{3}} = \frac{4}{5}
\]
Thus, the coordinates of point \( R \) are:
\[
R = (0, \frac{19}{5}, \frac{4}{5})
\]
### Final Answer
The ratio in which the line segment \( PQ \) is divided by the \( yz \)-plane is \( 2:3 \) and the point of contact is \( R(0, \frac{19}{5}, \frac{4}{5}) \).
