Find the equation of the plane which bisects the line segment joining the points `A(2,3,4)` and `B(4,5,8)` at right angles.

Options

A. `x+y+2z=19`
B. `x+y-2z=19`
C. `x+y-2z=38`
D. `x+y+2z=38`

Answer

A

Explanation

Since the equation of a plane is bisecting perpendicular the line joining the points `A(2,3,4)` and `B(4,5,8)` at right angles
So,mid pont of `AB` is `((2+4),2,(3+5),2,(4+8)/2)` i.e. `(3,4,5)`
Also `barN=(4-2)hati+(5-3)hatj+(B-4)hatk=2hatk+2hatj+4hatk`
So, the required equation of the planes is `(vecr-veca).vecN=0`
`implies[(x-3)hati+(y-4)hatj+(z-6)hatk].(2hati+2hatj+4hatk)=0 [ :' veca=3hati+4hatj+6hatk]`
`implies 2x-6+2y-8+4z-24=0`
`implies 2x+2y+4z=38`
`:. x+y+2z=19`