Question

Write the cartesian equation of the plane bisecting the line segment joining the points a (2, 3, 5) and b (4, 5, 7) at right angles.

Answer 1

x + y + z = 13

Since the plane bisects the line-segment at right angles, the vector from either of the points to the other must be perpendicular to the plane.

This shall give us a set of direction ratios for the normal to the plane:

(l, m, n) = (4 - 2, 5 - 3, 7 - 5) = (2, 2, 2)

Note: l, m, n are direction ratios, not cosines.

Now, the mid-point of this segment shall be a point through which the plane definitely passes, so:

(3, 4, 6) is a point on the plane.

Hence, the Cartesian equation of the plane becomes:

2(x - 3) + 2(y - 4) + 2(z - 6) = 0

Or, x + y + z - 3 - 4 - 6 = 0.

So, x + y + z = 13.