The equation of the plane which is perpendicular bisector of the line joining the points `A(1, 2, 3)` and `B(3, 4, 5)` is

Options

A. `x+y+z=9`
B. `x+y+z=-9`
C. `2xx3y+4z=9`
D. `2x+3y+4z=-9`

Explanation

To find the equation of the plane that is the perpendicular bisector of the line segment joining the points \( A(1, 2, 3) \) and \( B(3, 4, 5) \), we can follow these steps:

### Step 1: Find the Midpoint of the Line Segment AB
The midpoint \( C \) of the line segment joining points \( A \) and \( B \) 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 \( A \) and \( B \):
\[ C\left( \frac{1 + 3}{2}, \frac{2 + 4}{2}, \frac{3 + 5}{2} \right) = C\left( \frac{4}{2}, \frac{6}{2}, \frac{8}{2} \right) = C(2, 3, 4) \]

### Step 2: Find the Direction Vector AB
The direction vector \( \overrightarrow{AB} \) can be found by subtracting the coordinates of point \( A \) from point \( B \):
\[ \overrightarrow{AB} = B - A = (3 - 1, 4 - 2, 5 - 3) = (2, 2, 2) \]

### Step 3: Determine the Normal Vector
The normal vector \( \mathbf{n} \) of the plane is the same as the direction vector \( \overrightarrow{AB} \):
\[ \mathbf{n} = (2, 2, 2) \]

### Step 4: Use the Point-Normal Form of the Plane Equation
The equation of a plane can be expressed in the point-normal form:
\[ \mathbf{n} \cdot (\mathbf{r} - \mathbf{r_0}) = 0 \]
Where \( \mathbf{r} = (x, y, z) \) is a general point on the plane, and \( \mathbf{r_0} = (2, 3, 4) \) is a point on the plane. Substituting the values:
\[ (2, 2, 2) \cdot ((x, y, z) - (2, 3, 4)) = 0 \]
This expands to:
\[ (2, 2, 2) \cdot (x - 2, y - 3, z - 4) = 0 \]
Calculating the dot product:
\[ 2(x - 2) + 2(y - 3) + 2(z - 4) = 0 \]
Simplifying:
\[ 2x - 4 + 2y - 6 + 2z - 8 = 0 \]
\[ 2x + 2y + 2z - 18 = 0 \]
Dividing through by 2:
\[ x + y + z = 9 \]

### Final Equation of the Plane
Thus, the equation of the plane which is the perpendicular bisector of the line segment joining points \( A \) and \( B \) is:
\[ x + y + z = 9 \]