a p=(x,y,z)f=(c,0,0) , f'=(-c,0,0)||||p-f||-||p-f'||||=2a||||(x-c,y,z)||-||(x+c,y,z)||||=2a|(x-c)2+y2+z2-(x+c)2+y2+z2|=2a((x-c)2+y2+z2-(x+c)2+y2+z2)2=4a2(x-c)2+y2+z2-2((x-c)2+y2+z2)((x+c)2+y2+z2)+(x+c)2+y2+z2=4a22x2+2y2+2z2+2c2-4a2=2((x-c)2+(y2+z2))((x+c)2+(y2+z2))x2+y2+z2+c2-2a2=((x-c)2+(y2+z2))((x+c)2+(y2+z2)) p2=x2+y2+z2c2-2a2≡Ay2+z2≡Bp2+A=((x-c)2+B)((x+c)2+B)p4+2Ap2+A2=(x-c)2(x+c)2+B((x-c)2+(x+c)2)+B2p4+2Ap2+A2=((x+c)(x-c))2+2B(x2+c2)+B2p4+2Ap2+A2=(x2-c2)2+2B(x2+c2)+B2 (x2+y2+z2)2+2(c2-2a2)(x2+y2+z2)+(c2-2a2)2=x4-2c2x2+c4+2(y2+z2)(x2+c2)+(y2+z2)2 x4+y4+z4+2(x2y2+y2z2+z2x2)+2(c2-2a2)(x2+y2+z2)+c4-4a2c2+4a4=x4-2c2x2+c4+2(x2y2+z2x2)+2c2(y2+z2)+y4+2y2z2+z4 x4+y4+z4+2(x2y2+y2z2+z2x2)+2c2(x2+y2+z2)-4a2(x2+y2+z2)+c4-4a2c2+4a4=x4+y4+z4+2(x2y2+y2z2+z2x2)+2c2(-x2+y2+z2)+c4 2c2x2-4a2(x2+y2+z2)-4a2c2+4a4=-2c2x2 4c2x2-4a2(x2+y2+z2)-4a2c2+4a4=0c2x2

a2-(x2+y2+z2)-c2+a2=0(c2-a2)x2

a2-y2-z2=c2-a2x2

a2-y2

c2-a2-z2

c2-a2=1b2≡c2-a2 x2

a2-y2

b2-z2

b2=1