YOZ-plane divides the join of A (3, 5, 6) and B ( 4, 6, - 3)

Options

A. internally in the ratio 3 : 4
B. internally in the ratio 4: 3
C. externally in the ratio 3 : 4
D. externally in the ratio 4 : 3

Answer

To solve the problem of how the YOZ-plane divides the line segment joining points A(3, 5, 6) and B(4, 6, -3), we will follow these steps:

### Step 1: Understand the problem
The YOZ-plane is defined as the plane where the x-coordinate is 0. We need to find the point where the line segment joining points A and B intersects this plane and determine the ratio in which it divides the segment.

### Step 2: Set up the coordinates
Let:
- A = (3, 5, 6)
- B = (4, 6, -3)

### Step 3: Use the section formula
The section formula states that if a point P divides the line segment joining points A(x1, y1, z1) and B(x2, y2, z2) in the ratio λ:μ, the coordinates of point P can be given by:
\[ P\left(\frac{\lambda x_2 + \mu x_1}{\lambda + \mu}, \frac{\lambda y_2 + \mu y_1}{\lambda + \mu}, \frac{\lambda z_2 + \mu z_1}{\lambda + \mu}\right) \]

### Step 4: Set the x-coordinate to 0
Since the point lies on the YOZ-plane, we set the x-coordinate of P to 0:
\[ 0 = \frac{\lambda \cdot 4 + \mu \cdot 3}{\lambda + \mu} \]

### Step 5: Solve for the ratio
From the equation above, we can multiply both sides by (λ + μ):
\[ 0 = \lambda \cdot 4 + \mu \cdot 3 \]
Rearranging gives:
\[ \lambda \cdot 4 = -\mu \cdot 3 \]
This can be rewritten as:
\[ \frac{\lambda}{\mu} = -\frac{3}{4} \]

### Step 6: Express the ratio
Let’s express the ratio λ:μ:
\[ \lambda : \mu = -3 : 4 \]
This indicates that the division is external since the ratio is negative.

### Step 7: Write the final answer
Thus, the YOZ-plane divides the line segment joining points A and B externally in the ratio 3:4.