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In an Agrad plane z(1),z(2) and z(3) ar...

In an Agrad plane `z_(1),z_(2) and z_(3)` are, respectively, the vertices of an isosceles trinagle ABC with AC= BC and `/_CAB = theta`. If `z_(4)` is incentre of triangle, then
The value of `AB xx AC// (IA)^(2)` is

A

`((z_(2)-z_(1))(z_(3)-z_(1)))/((z_(4) -z_(1))^2)`

B

`((z_(2) -z_(1))(z_(1)-z_(3)))/((z_(4)-z_(1))^(2))`

C

`((z_(4)-z_(1))^(2))/((z_(2) -z_(1))(z_(3)-z_(1)))`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A

`(ABxxAC)/((IA)^(2)) = (AB)/(IA) xx (AC)/(IA)`
`/_AB = (theta)/(2),/_IAC = (theta)/(2)`
`(z_(2) -z_(1))/(z_(4) -z_(1)) =(|z_(2) -z_(1)|)/(|z_(4) -z_(1)|) e^(-(itheta)/(2)`
` and (z_(3)-z_(1))/(z_(4) -z_(1)) (z_(3)-z_(1))/(z_(4) -z_(1)) = (|z_(2)-z_(1)|)/(|z_(4)-z_(1)|) (| z_(3)-z_(1)|)/(|z_(4) -z_(1)|)`
` rArr ((z_(2) -z_(1))(z_(3)-z_(1)))/((z_(4)-z_(1))^(2)) = (AB xxAC)/(IA^(2))" "(1)`
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