Find the equation of the locus of point P, the sum of whose distances from the points
A(4, 0, 0) and B(0, -4, 0) is 5.

Explanation

Let the point P`(alpha, beta, gamma)`, then PA+ PB = 5
`rArr sqrt((alpha - 4)^(2) + beta^(2) +gamma ^(2))+ sqrt(alpha^(2) +(beta+4)^(2)+gamma^(2))=5`
`rArr sqrt((alpha - 4)^(2) + beta^(2) +gamma ^(2))+ =5-sqrt(alpha^(2) +(beta+4)^(2)+gamma^(2))`
On squaring both sides, we yet
`alpha^(2)+ beta^(2) +gamma^(2) - 8alpha + 16= 25 + alpha^(2)+beta ^(2) + gamma^(2) + 8beta + 16 -10sqrt(alpha^(2)+(beta+4)^(2)+gamma^(2))`
`rArr 10sqrt(alpha^(2)+(beta+4)^(2)+gamma^(2))=25+8alpha+8beta`
Again, squaring both sides, we get
`100[alpha^(2)+beta^(2)+gamma^(2)+ 8beta + 16] = 625 + 64alpha^(2)+64beta^(2)+400alpha+400beta +128alpha beta`
`rArr 36alpha^(2) +36beta^(2)+100gamma^(2)-400alpha+400beta-128alphabeta+975=0`
Hence, the locus of the point P is
`36x^(2)+36y^(2)+100z^(2)-128xy-400x+400y+975=0`