<p>A plane bisects the line segment joining the points \((1, 2, 3)\) and \((-3, 4, 5)\) at right angles. Then this plane also passes through the point</p>
Step-by-Step Solution
Key Concept: A plane that perpendicularly bisects a line segment passes through its midpoint and has a normal vector equal to the direction vector of the line segment.
Step 1: Find the midpoint of the line segment Midpoint M = ((1-3)/2, (2+4)/2, (3+5)/2) = (-1, 3, 4) Step 2: Find the direction vector of the line segment Direction vector = (-3-1, 4-2, 5-3) = (-4, 2, 2) Step 3: Write the equation of the perpendicular bisecting plane The normal vector to the plane is (-4, 2, 2) or simplified (-2, 1, 1) Plane equation: -2(x+1) + 1(y-3) + 1(z-4) = 0 -2x - 2 + y - 3 + z - 4 = 0 Step 4: Simplify -2x + y + z = 9 Step 5: Check which point satisfies this equation Substitute the given options into -2x + y + z = 9 to find the point that lies on the plane. ∴ Answer: B
Correct Answer: B