Question

Harmonic conjugate of (2,3,4) with respect to (3,-2,-2) and (6,-17,-4)

Answer 1

3*6=18
17*2=34
4*2=8
18,34,8

Answer 2

The harmonic congugate is $(\dfrac{18}{5},-5,\dfrac{4}{5}).$

Step-by-step explanation:

Let P(2, 3, 4), A(3, - 2, 2) and B(6,-17,-4)

Let P divides AB in the ratio k : 1, then

$2=\dfrac{6k+3}{k+1}$

⇒ 2k + 2 = 6k + 3

⇒ 6k - 2 k = 2 - 3

⇒ 4k = - 1

⇒ $k=-\dfrac{1}{4}$

Harmonic congugate Q divides in the ratio - k : 1 i.e., $\dfrac{1}{4}:1$

$Q=(\dfrac{6+12}{5},\dfrac{-17-8}{5},\dfrac{-4+8}{5} )$

$=(\dfrac{18}{5},\dfrac{-25}{5},\dfrac{4}{5})$

$=(\dfrac{18}{5},-5,\dfrac{4}{5})$

Hence, the harmonic congugate is $(\dfrac{18}{5},-5,\dfrac{4}{5}).$