Find the equation of the set of points P, the sum of whose distances from A(4,0,0) and B(−4,0,0) is equal to 10.
Let the coordinates of P be (x,y,z).
The coordinates of points A and B are (4,0,0) and (−4,0,0) respectively.
It is given that PA+PB=10
⇒√(x−4)2+y2+z2+√(x+4)2+y2+z2=10
⇒√(x−4)2+y2+z2=10−√(x+4)2+y2+z2
On squaring both sides, we obtain
⇒(x−4)2+y2+z2=100+(x+4)2+y2+z2−20√(x+4)2+y2+z2
⇒x2−8x+16+y2+z2=100−20√x2+8x+16+y2+z2+x2+8x+16+y2+z2
⇒20√x2+8x+16+y2+z2=100+16x
⇒5√x2+8x+16+y2+z2=(25+4x)
On squaring both sides again, we obtain
25(x2+8x+16+y2+z2)=625+16x2+200x
⇒25x2+200x+400+25y2+25z2=625+16x2+200x
⇒9x2+25y2+25z2−225=0
Thus the required equation is 9x2+25y2+25z2−225=0.