If the points `(1,1,p)` and `(-3,0,1)` be equidistant from the plane `vecr.(3hati+4hatj-12hatk)+13`, find the values of p.

Explanation

The eqation of the given plane is Cartesian form is
`xhati+yhatj+zhatk. (3hati+4hatj-12hatk)+13=0`.
`rArr 3x+4y-12z+13=0`. …………(i)
It is being given that the distances of the plane (i) from each of the points A(1,1,p) and B(-3,0,1) are equal.
`therefore (|(3xx1)+(4xx1)-(12xxp)+13|)/sqrt(3^(2)+4^(2)+(-12)^(2))=|(3xx(-3)+(4xx0)-(12xx1))|/sqrt(3^(2)+4^(2)+(-12)^(2))`
`|20-12p|=|-8| rArr |20-12p|=8`
`rArr (20-12p)=8` or `-(20-12p)=8`
`rArr 12p=2` or `12p=28`
`rArr p=1` or `p=7/3`.
Hence, `p=1` or `p=7/3`.