Question

Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, -3, 1) and (2, 3, -5) at the right angles. If a, b, c, d are integers, then the minimum value of (a² + b² + c² + d²) is __________.

Hint:

For a plane bisecting a line segment $PQ$ at right angles, the normal vector is $\vec{PQ}$ and it passes through the midpoint of $PQ$.

Answer 1

Correct Answer: 28

Step 1: The plane is the perpendicular bisector.
Step 2: Direction ratios of the normal to the plane = $(4-2, -3-3, 1-(-5)) = (2, -6, 6) \parallel (1, -3, 3)$.
Step 3: Midpoint of the line segment = $\left(\frac{4+2}{2}, \frac{-3+3}{2}, \frac{1-5}{2}\right) = (3, 0, -2)$.
Step 4: Equation of the plane: $1(x-3) - 3(y-0) + 3(z+2) = 0$.
Step 5: $x - 3y + 3z + 3 = 0$.
Step 6: $a=1, b=-3, c=3, d=3$.
Step 7: $a^2 + b^2 + c^2 + d^2 = 1 + 9 + 9 + 9 = 28$.