Question

If coordinate of A and B is (0,0) and (9,0) respectively then find point which divide ab externally in 1 ratio 2, and find its harmonic conjugate also.​

Answer 1

Answer:

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Answer 2

• The point which divide AB externally in the ratio 1 : 2 is ( - 9,0)

• The harmonic conjugate of the point is (3,0)

Given :

The coordinate of A and B is (0,0) and (9,0) respectively

To find :

• The point which divide AB externally in the ratio 1 : 2

• The harmonic conjugate of the point

Formula Used :

1. The coordinates of the point where the line joining the points (x₁ , y₁) & (x₂ , y₂) is divided Internally in the ratio m : n

$= \displaystyle\sf{ \bigg( \frac{mx_2+nx_1 \: }{m + n} \: \: , \: \frac{my_2+ny_1 \: }{m + n} \bigg) }$

2. If a point P divides AB in the ratio m : n externally then harmonic conjugate divides AB internally in the ratio m : n

Solution :

Step 1 of 3 :

Write down the given points

Here the given points are A(0,0) , B(9,0)

Step 2 of 3 :

Find the point which divide AB externally in the ratio 1 : 2

We know that , the coordinates of the point where the line joining the points (x₁ , y₁) & (x₂ , y₂) is divided externally in the ratio m : n is

$\displaystyle\sf{ \bigg( \frac{mx_2 - nx_1 \: }{m - n} \: \: , \: \frac{my_2 - ny_1 \: }{m - n} \bigg) }$

∴ The coordinate of the point which divide AB externally in the ratio 1 : 2

$\displaystyle\sf{ = \bigg( \frac{1 \times 9 - 2 \times 0 \: }{1 - 2} \: \: , \: \frac{1 \times 0 - 2 \times 0 \: }{1 - 2} \bigg) }$

$\displaystyle\sf{ = \bigg( \frac{ 9 - 0 \: }{ - 1 } \: \: , \: \frac{ 0 }{ - 1 } \bigg) }$

$\displaystyle\sf{ =( - 9 , 0) }$

Step 3 of 3 :

Find the harmonic conjugate of the point

Since the point which divide AB externally in the ratio 1 : 2

∴ The harmonic conjugate of the point will divide AB internally in the ratio 1 : 2

Hence harmonic conjugate of the point

$\displaystyle\sf{ = \bigg( \frac{1 \times 9 + 2 \times 0 \: }{1 + 2} \: \: , \: \frac{1 \times 0 + 2 \times 0 \: }{1 + 2} \bigg) }$

$\displaystyle\sf{ = \bigg( \frac{ 9 + 0 \: }{ 3 } \: \: , \: \frac{ 0 }{ 3 } \bigg) }$

$\displaystyle\sf{ =(3 , 0) }$

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