Question

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 ?

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

Answer 1

The Correct Option is B

1. First, let's find the midpoint of the line segment joining the points \((-3, -3, 4)\) and \( (3, 7, 6) \). This midpoint is given by the formula: 

\(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right)\)

Plugging in the values, we get: \(\left( \frac{-3 + 3}{2}, \frac{-3 + 7}{2}, \frac{4 + 6}{2} \right)\)= \((0, 2, 5)\).

1. Next, we determine the direction vector of the line segment. The direction vector is given by subtracting the coordinates of the two points:

\((-3, -3, 4) \rightarrow (3, 7, 6)\) gives the vector \((3 - (-3), 7 - (-3), 6 - 4)\) = \((6, 10, 2)\).

1. The plane that bisects the line segment at right angles has its normal vector parallel to the direction vector of the line segment. Therefore, the normal vector of the plane is \((6, 10, 2)\).
2. The equation of the plane can be written as:

\(6(x - 0) + 10(y - 2) + 2(z - 5) = 0\).

1. Simplifying, we obtain:

\(6x + 10y + 2z = 0 + 20 + 10 \implies 6x + 10y + 2z = 30\)

1. Thus, the equation of the plane is \(6x + 10y + 2z = 30\).
2. We now test each point to see if it satisfies the plane equation.
   • For point \((4, -1, 7)\):

\(6(4) + 10(-1) + 2(7) = 24 - 10 + 14 = 28 \). Not equal to 30, so not on the plane.

• For point \((4, 1, -2)\):

\(6(4) + 10(1) + 2(-2) = 24 + 10 - 4 = 30 \). Equal to 30, so this point lies on the plane.

• For point \((-2, 3, 5)\):

\(6(-2) + 10(3) + 2(5) = -12 + 30 + 10 = 28 \). Not equal to 30, so not on the plane.

• For point \((2, 1, 3)\):

\(6(2) + 10(1) + 2(3) = 12 + 10 + 6 = 28 \). Not equal to 30, so not on the plane.

1. Hence, the correct answer is point \((4, 1, -2)\), which satisfies the equation of the bisecting plane.