Find the locus of `P\ if\ P A^2+P B^2=2k^2w h e r e\ A\ a n d\ B` are points (3,4,5) and (-1,3,7).

Explanation

given that the points are `(3,4,5) and (-1,3,7)`
we have to find the locus of `P\ if\ P A^2+P B^2=2k^2`
Let the required point `P(x, y, z) `
The distance between P and A is ,
`PA=sqrt((x-3)^2+(y-4)^2+(z-5)^2)`
The distance between P and B is ,
`PB=sqrt((x+1)^2+(y-3)^2+(z-7)^2)`
According to question `P A^2+P B^2=2k^2`
`(x-3)^2+(y-4)^2+(z-5)^2+(x+1)^2+(y-3)^2+(z-7)^2=2k^2`
`=>x^2+9-6x+y^2+16-8y+z^2+25-10z+x^2+1+2x+y^2+9-6y+z^2+49+14z=2k^2`
`=>2x^2+2y^2+2z^2-4x-14y+4z+109=2k^2`
`=>2x^2+2y^2+2z^2-4x-14y+4z+109-2k^2=0`
Hence locus is `2x^2+2y^2+2z^2-4x-14y+4z+109-2k^2=0`