The ratio in which the line segment joining the point A (4, 8, 10) and B (6, 10,- 8) is divided by yz - plane is

Options

A. 2 : 3 internally
B. 2 : 3 externally
C. 3 : 2 internally
D. 3 : 2 externally

Answer

To find the ratio in which the line segment joining the points A(4, 8, 10) and B(6, 10, -8) 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 any point on the yz-plane will have its x-coordinate equal to 0.

### Step 2: Use the section formula
Let the point of intersection of the line segment AB with the yz-plane be P(0, y, z). According to the section formula, if a point P divides the line segment joining points A(x1, y1, z1) and B(x2, y2, z2) in the ratio k:1, then the coordinates of P can be given by:

\[ P = \left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1}, \frac{kz_2 + z_1}{k + 1} \right) \]

### Step 3: Set up the equations
For our points A(4, 8, 10) and B(6, 10, -8), we can substitute into the section formula:

\[ 0 = \frac{k \cdot 6 + 4}{k + 1} \]
This equation represents the x-coordinate of point P.

### Step 4: Solve for k
To solve for k, we set the numerator equal to zero:

\[ k \cdot 6 + 4 = 0 \]

Solving this gives:

\[ 6k + 4 = 0 \implies 6k = -4 \implies k = -\frac{4}{6} = -\frac{2}{3} \]

### Step 5: Interpret the ratio
The ratio in which the line segment is divided is k:1. Since k = -2/3, the ratio is:

\[ \text{Ratio} = \left| -\frac{2}{3} \right| : 1 = 2 : 3 \]

Since k is negative, it indicates that the division is external.

### Final Answer
The ratio in which the line segment joining points A and B is divided by the yz-plane is 2:3 externally.

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