Question

find the harmonic conjugate of (7,5) with respect to (4,2) (9,7)​

Answer 1

LOGIN

JOIN NOW



Type your question...



MATHS

The harmonic conjugate of (4,1) with respect to the points (3,2) and (-1,6) is

A .

(−4,−1)

B .

(1,4)

C .

(67​,68​​)

D .

(37​,38​​)

December 20, 2019Sakalabhaktula Behl

Share

Save

ANSWER

Let (4,1) divide (3,2) and (−1,6) in k:1


⇒ 4=k+1−k+3​⇒4k+4=−k+3⇒5k=−1⇒k=5−1​


⇒ it divide them in 1:5 externally 


its harmonic conjugate (P) would divide them in 1:5 internally


⇒Px​=6−1+15​=614​=37​,  


Py​=66+10​=616​=38​⇒(D

Answer 2

The correct answer is (19, 11).

Given: Points (4, 2) and (9, 7).

To Find: The harmonic conjugate (7, 5).

Solution:

Let say that the ratio is m:n.

$x=\frac{m*x_{2 }+n*x_{1} }{m+n}$

$7=\frac{9*m+4*n}{m+n}$

7m + 7n = 9m + 4n

3n = 2m  (equation 1)

$y=\frac{m*y_{2 }+n*y_{1} }{m+n}$

$5=\frac{7*m+2*n}{m+n}$

5m + 5n = 7m + 2n

3n = 2m  (equation 2)

From both the equations we are getting

$\frac{m}{n}=\frac{3}{2}$

The division is internal as in equation we are getting positive sign.

Point A = (4, 2)

Point B = (9, 7)

Ratio = 3: 2

$x=\frac{m*x_{2 }-n*x_{1} }{m-n}$

$=\frac{9*3-4*2}{3-2}$

x = 19

$y=\frac{m*y_{2 }-n*y_{1} }{m-n}$

= $=\frac{7*3-5*2}{3-2}$

y = 11

Hence, the harmonic conjugate is (19, 11).

#SPJ3