Question

The harmonic conjugate of (4,-2) with respect to the points (2, -4) and (7,1) is

Answer 1

Greetings, 
The answer to your question is typed below↓
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Let A(2, -4) and B(7,1)
The ratio in which (4,-2) divides |AB| is
(2-4):(4-1) = -2 : 3

∴ The point which divides AB in the ratio -2 : 3 is

P(x,y)=[ {-2(7) - 3(2)}/3-2 , {-2(1) - 3(-4)}/3-2]
∴ P(x,y) = (-20,10)           [Ans.]
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Enjoy ;)

Answer 2

Given :  points (2, -4) and (7,1)

To find : harmonic conjugate of (4,-2)

Solution:

First find in which ratio  (4,-2) Divides  (2, -4) and (7,1)

let say m : n ratio

then    4  = ( m*7 + n*2)/(m + n)     or   -2  = (m *(1) + n*(-4)/(m + n)

=> 4m + 4n = 7m + 2n     or  -2m - 2n = m - 4n

=> 3m = 2n                       or  2n  = 3m

=> m/n = 2/3

The harmonic conjugate of (4,-2) with respect to the points (2, -4) and (7,1)

will divide point in -2 : 3 ratio

= ( -2*7 + 3*2)/(-2 + 3)  ,   (-2 *(1) + 3*(-4))/(-2 + 3)

= -8  , -14

( -8  , -14) is  harmonic conjugate of (4,-2) with respect to the points (2, -4) and (7,1)

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