The plane which bisects the line segment joining the points (-3, -3, 4) and (3, 7, 6) at right angles, passes through which one of the following points?

Options

A. `(4, -1,7)`
B. `(2,1,3)`
C. `(-2,3,5)`
D. `(4,1,-2)`

Answer

To solve the problem, we need to find the equation of the plane that bisects the line segment joining the points A(-3, -3, 4) and B(3, 7, 6) at right angles. Then, we will determine which of the given points lies on this plane.

### Step 1: Find the Midpoint of the Line Segment AB
The midpoint M of the line segment joining points A and B can be calculated using the midpoint 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 A and B:
\[ M = \left( \frac{-3 + 3}{2}, \frac{-3 + 7}{2}, \frac{4 + 6}{2} \right) = \left( 0, 2, 5 \right) \]

### Step 2: Find the Direction Ratios of Line Segment AB
The direction ratios of the line segment AB can be found by subtracting the coordinates of A from B:
\[ \text{Direction Ratios} = (3 - (-3), 7 - (-3), 6 - 4) = (6, 10, 2) \]

### Step 3: Write the Equation of the Plane
The equation of a plane can be expressed in the form:
\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]
where (x0, y0, z0) is a point on the plane (which is M in this case) and (a, b, c) are the direction ratios of the normal to the plane (which are the same as the direction ratios of AB).

Substituting the values:
\[ 6(x - 0) + 10(y - 2) + 2(z - 5) = 0 \]
Expanding this gives:
\[ 6x + 10y - 20 + 2z - 10 = 0 \]
Simplifying:
\[ 6x + 10y + 2z - 30 = 0 \]
Dividing through by 2:
\[ 3x + 5y + z = 15 \]

### Step 4: Check Which Given Point Lies on the Plane
Now, we need to check which of the given points satisfies the equation \(3x + 5y + z = 15\).

Assuming we have a point (4, -2, 12) to check:
\[ 3(4) + 5(-2) + 12 = 12 - 10 + 12 = 14 \quad \text{(not equal to 15)} \]

Assuming we have another point (0, 2, 5) to check:
\[ 3(0) + 5(2) + 5 = 0 + 10 + 5 = 15 \quad \text{(equal to 15)} \]

Thus, the point (0, 2, 5) lies on the plane.

### Final Answer
The plane which bisects the line segment joining the points (-3, -3, 4) and (3, 7, 6) at right angles passes through the point (0, 2, 5).

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