Question

The harmonic conjugate of (-9,27) with respect to the points (1,7) and (6,-3)is

Answer 1

(3 , 3) is The harmonic conjugate of (-9,27) with respect to the points (1,7) and (6,-3)

Step-by-step explanation:

The harmonic conjugate of (-9,27) with respect to the points (1,7) and (6,-3)i

First find in which ration  (-9 , 27)  Divides  (1 , 7)  & (6 , - 3)

let say m : n ratio

then -9  = ( m*6 + n*1)/(m + n)     or   -27 = (m *(-3) + n*7)/(m + n)

=> -9m - 9n = 6m + n      

=> -15m = 10n                  

=> m/n = -10/15

=> m/n = - 2/3

=> m:n = - 2: 3

The harmonic conjugate of (-9,27) with respect to the points (1,7) and (6,-3)

will divide point in 2 : 3 ratio

= ( 2*6 + 3*1)/(2 + 3)  ,   (2 *(-3) + 3*7)/(2 + 3)

= 15/5  , 15/5

= 3 , 3

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Answer 2

The harmonic conjugate of (-9,27) with respect to the points (1,7) and (6,-3) is (3, 3).

To find : The harmonic conjugate of ( -9, 27 ) with respect to the points ( 1, 7) and ( 6, -3 ) is = ?

Given data :

Let the points a(1, 7) and b(6, -3) and the center p(-9, 27).

$x_1$ = 1    ;     $x_2$ = 6    ;   $y_1$ = 7   ;    $y_2$ = -3    

Let a : b be the ratio ⇒ a : b = p

Harmonic conjugate :

     $\frac{a\times\\(x_2)+b\times\\(x_1)}{a+b} = \frac{a\times\\(6)+b\times\\(1)}{a+b}= -9$    -----> (1)

                                  (or)  

   $\frac{a\times\\(y_2)+b\times\\(y_1)}{a+b} = \frac{a\times\\(7)+b\times\\(-3)}{a+b}= 27$   ------> (2)

Taking values of :

$\frac{a\times\\(6)+b\times\\(1)}{a+b}= -9$

$\frac{6 a + b}{a+b} = -9$

6 a + b = -9 ( a + b )

6 a + b = -9 a - 9 b

Arrange the "a" terms and "b" terms like,

6 a + 9 a = -9 b - b

15 a = -10 b

$\frac{a}{b} = -\frac{10}{15}$

$\frac{a}{b} = -\frac{2}{3}$

a : b = 2 : 3

Hence, the value of a and b is 2 and 3.

Substituting the value of a and b in the above equation (1) and (2), we get

(1)    $\frac{a\times\\(6)+b\times\\(1)}{a+b}= -9$

       $\frac{6 a + b}{a+b}$

      $\frac{6\times2 + 3}{2+3}$

     $\frac{12+3}{5} = \frac{15}{5} = 3$

(2)  $\frac{a\times\\(-3)+b\times\\(7)}{a+b}$

    $\frac{2\times\\(-3)+3\times\\(7)}{2+3}$

   $\frac{-6+21}{5}=\frac{15}{5}=3$

Therefore, the point is (3, 3).

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