Find the ratio in which line segment joining the points (2,4,5) and (3,5,-9) is divided by yz -plane.

Answer

To find the ratio in which the line segment joining the points (2, 4, 5) and (3, 5, -9) is divided by the YZ-plane, we can follow these steps:

### Step 1: Understand the YZ-plane
The YZ-plane is defined by the equation \( x = 0 \). This means that we need to find the point on the line segment between the two given points where the x-coordinate is zero.

### Step 2: Use the section formula
Let the points be:
- \( P_1(2, 4, 5) \) (denoted as \( (x_1, y_1, z_1) \))
- \( P_2(3, 5, -9) \) (denoted as \( (x_2, y_2, z_2) \))

We can use the section formula to find the coordinates of the point that divides the line segment joining \( P_1 \) and \( P_2 \) in the ratio \( m:n \):
\[ x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}, \quad z = \frac{mz_2 + nz_1}{m+n} \]

### Step 3: Set up the equation for x-coordinate
Since we want the x-coordinate to be zero (as it lies on the YZ-plane), we can set up the equation:
\[ 0 = \frac{m \cdot 3 + n \cdot 2}{m+n} \]

### Step 4: Solve for the ratio
From the equation above, we can simplify it:
\[ m \cdot 3 + n \cdot 2 = 0 \]
This can be rearranged to:
\[ 3m + 2n = 0 \]
From this equation, we can express \( m \) in terms of \( n \):
\[ 3m = -2n \quad \Rightarrow \quad \frac{m}{n} = -\frac{2}{3} \]
This indicates that the ratio \( m:n \) is \( -2:3 \).

### Step 5: Interpret the ratio
The negative sign indicates that the division is external. Therefore, the ratio in which the line segment joining the points (2, 4, 5) and (3, 5, -9) is divided by the YZ-plane is \( 2:3 \) externally.

### Final Answer
The ratio in which the line segment is divided by the YZ-plane is \( 2:3 \) externally.

---