The equation of the plane, which bisects the line joining the points (1, 2, 3) and (3, 4, 5) at right angles, is

Options

A. `x+y+z=0`
B. `x+y-z=0`
C. `x+y+z=9`
D. `x+y-z+9=0`

Answer

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

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

### Step 2: Find the Direction Ratios of the Line Segment
The direction ratios of the line segment joining points \( A \) and \( B \) can be found by subtracting the coordinates of \( A \) from \( B \):
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) = (3 - 1, 4 - 2, 5 - 3) = (2, 2, 2) \]

### Step 3: Find the Normal to the Plane
Since the plane bisects the line segment at right angles, the direction ratios of the line segment will be the normal to the plane. Therefore, the normal direction ratios \( (A, B, C) \) of the plane are:
\[ (2, 2, 2) \]

### Step 4: Write the Equation of the Plane
The general equation of a plane with normal direction ratios \( (A, B, C) \) passing through a point \( (x_1, y_1, z_1) \) is given by:
\[ A(x - x_1) + B(y - y_1) + C(z - z_1) = 0 \]
Substituting \( A = 2, B = 2, C = 2 \) and the coordinates of the midpoint \( M(2, 3, 4) \):
\[ 2(x - 2) + 2(y - 3) + 2(z - 4) = 0 \]
Simplifying this equation:
\[ 2x - 4 + 2y - 6 + 2z - 8 = 0 \]
\[ 2x + 2y + 2z - 18 = 0 \]
Dividing the entire equation by 2:
\[ x + y + z - 9 = 0 \]
Thus, the equation of the plane is:
\[ x + y + z = 9 \]

### Final Answer
The equation of the plane that bisects the line joining the points \( (1, 2, 3) \) and \( (3, 4, 5) \) at right angles is:
\[ x + y + z = 9 \]