The equation of the plane which bisects the line joining `(2, 3, 4)` and `(6,7,8)`

Options

A. `x+y+z+15=0`
B. `x-y-z-15=0`
C. `x-y+z-15=0`
D. `x+y+z-15=0`

Explanation

To find the equation of the plane that bisects the line joining the points \( A(2, 3, 4) \) and \( B(6, 7, 8) \), we can follow these steps:

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( C \) of the line segment joining points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) can be calculated using the midpoint formula:
\[ C\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
Substituting the coordinates of points \( A(2, 3, 4) \) and \( B(6, 7, 8) \):
\[ C\left( \frac{2 + 6}{2}, \frac{3 + 7}{2}, \frac{4 + 8}{2} \right) = C(4, 5, 6) \]

### Step 2: Find the Direction Ratios of the Line
The direction ratios of the line joining points \( A \) and \( B \) can be found by subtracting the coordinates:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (6 - 2, 7 - 3, 8 - 4) = (4, 4, 4) \]
To find the direction cosines, we can normalize these direction ratios. The direction cosines are given by:
\[ \left( \frac{4}{\sqrt{4^2 + 4^2 + 4^2}}, \frac{4}{\sqrt{4^2 + 4^2 + 4^2}}, \frac{4}{\sqrt{4^2 + 4^2 + 4^2}} \right) \]
Calculating the magnitude:
\[ \sqrt{4^2 + 4^2 + 4^2} = \sqrt{48} = 4\sqrt{3} \]
Thus, the direction cosines are:
\[ \left( \frac{4}{4\sqrt{3}}, \frac{4}{4\sqrt{3}}, \frac{4}{4\sqrt{3}} \right) = \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \]

### Step 3: Write the Equation of the Plane
The general equation of a plane can be written as:
\[ ax + by + cz + d = 0 \]
Where \( (a, b, c) \) are the direction ratios. Here, we can take \( a = 1 \), \( b = 1 \), \( c = 1 \) (since they are proportional). Thus, the equation becomes:
\[ x + y + z + d = 0 \]

### Step 4: Substitute the Midpoint into the Plane Equation
Now we substitute the coordinates of the midpoint \( C(4, 5, 6) \) into the plane equation to find \( d \):
\[ 4 + 5 + 6 + d = 0 \implies 15 + d = 0 \implies d = -15 \]

### Step 5: Final Equation of the Plane
Substituting \( d \) back into the plane equation:
\[ x + y + z - 15 = 0 \]
Thus, the equation of the plane that bisects the line joining the points \( (2, 3, 4) \) and \( (6, 7, 8) \) is:
\[ x + y + z = 15 \]