Let A=(3,−2,2) B=(6,−17,−4) P=(2,3,4) Let P divides AB in the ratio of m:n
P=[
mx2+nx1
m+n
,
my2+ny1
m+n
,
mz2+nz1
m+n
] (2,3,4)=[
m(6)+n(3)
m+n
,
m(−17)+n(−2)
m+n
m(−4)+n(2)
m+n
] (2,3,4)=[
6m+3n
m+n
,
−17m−2n
m+n
,
−4m+2n
m+n
]
6m+3n
m+n
=2 6m+3n=2m+2n 4m=−n
m
n
=−
1
4
∴m:n=−1:4 We know that, If point P divides a line segment in the ratio of m:n, then its harmonic conjugate will divide same segment in the ratio of −m:n ∴ Required ratio =−m:n=1:4 ∴ Required harmonic conjugate