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

Options

A. `r-(3hat(i)+4hat(j)+6hat(k))=(2hat(i)+2hat(j)+4hat(k))`
B. `r-(2hat(i)+2hat(j)+4hat(k))=(3hat(i)+4hat(j)+6hat(k))`
C. `[r-(3hat(i)+4hat(j)+6hat(k))]*(2hat(i)+2hat(j)+4hat(k))=0`
D. `[r-(2hat(i)+2hat(j)+4hat(k))*(3hat(i)+4hat(j)+6hat(k))]=0`

Answer

C

Explanation

Now direction ratios of AB
`n=AB=OB-OA `
`=(4hat(i)+5hat(j)+8hat(k))-(2hat(i)+3hat(j)+4hat(k))=2hat(i)+2hat(j)+4hat(k)` Since, plane bisects AB. So, it will pass through the mid - point of AB.
The coordinates of mid - point of AB are
`((4+2)/(2),(5+3)/(2),(8+4)/(2))i.e.(3,4,6)`.
The equation of the plane is given by
`(r-a)*(n=0)rArr[r-(3hat(i)+4hat(j)+6hat(k))]*(2hat(i)+2hat(j)+4hat(k))=0`