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.

If the points `(1,1,p)` and `(-3,0,1)` be equidistant from the plane `

1.	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.
Answer» 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`.

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