Question

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: 

(A) (−3, 2, 1) 

(B) (3, 2, 1) 

(C) (−1, 2, 3) 

(D) (1, 2, −3)

Answer 1

Answer is (A) (−3, 2, 1)

From the given data, we can plot the graph as shown in the following figure:

Plane bisects the line AB perpendicularly. Therefore, the midpoint of AB is

((-3 + 1)/2, (4 + 2)/2, (3 + 5)/2) = (-1, 3, 4)

Now, the direction ratio of AB is (−4, 2, 2). 

Therefore, equation of plane is given by

−4(x + 1) + 2(y – 3) + 2(z − 4) = 0

⇒ −4x – 4 + 2y – 6 + 2z – 8 = 0

⇒ −4x + 2y + 2z – 18 = 0

⇒ −2x + y + z = 9

Only (−3, 2, 1) satisfies the plane