The ratio in which the line joining the ponts (2,5,4) and (3,5,4) is divided by the YZ plane is

Options

A. (a) 3:2 internally
B. (b) 2:3 internally
C. (c) 3:2 externally
D. (d) 2:3 externally

Answer

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

### Step 1: Identify the coordinates of the points
The points given are:
- Point A: (2, 5, 4)
- Point B: (3, 5, 4)

### Step 2: Understand the YZ plane
The YZ plane is defined by the equation x = 0. This means we need to find the point on the line joining A and B where x = 0.

### Step 3: Set up the ratio
Let the ratio in which the YZ plane divides the line segment AB be k:1. According to the section formula, the coordinates of the point P that divides the line segment joining A and B in the ratio k:1 can be expressed as:
\[ P_x = \frac{(x_2 \cdot k + x_1 \cdot 1)}{k + 1} \]
Where \(x_1\) and \(x_2\) are the x-coordinates of points A and B, respectively.

### Step 4: Substitute the coordinates
Substituting the coordinates of points A and B into the formula:
\[ P_x = \frac{(3k + 2 \cdot 1)}{k + 1} \]
Since we want to find where this point lies on the YZ plane, we set \(P_x = 0\):
\[ 0 = \frac{(3k + 2)}{k + 1} \]

### Step 5: Solve for k
To solve for k, we set the numerator equal to zero:
\[ 3k + 2 = 0 \]
\[ 3k = -2 \]
\[ k = -\frac{2}{3} \]

### Step 6: Determine the ratio
The ratio in which the line is divided is \(k:1\). Since k is negative, we can express the ratio as:
\[ \text{Ratio} = -\frac{2}{3}:1 \]
To express this in positive terms, we can multiply through by -1:
\[ \text{Ratio} = 2:3 \]

### Final Answer
Thus, the ratio in which the line joining the points (2, 5, 4) and (3, 5, 4) is divided by the YZ plane is **2:3**.