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In the Argand plane Z1,Z2 and Z3 are res...

In the Argand plane `Z_1,Z_2` and `Z_3` are respectively the verticles of an isosceles triangle ABC with AC=BC and `angleCAB=theta`. If `I(Z_4)` is the incentre of triangle , then :
The value of `(Z_4-Z_1)^2 (1 + cos theta) sec theta` is :

A

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

B

`(z_(2)-z_(1))(z_(3) -z_(1))`

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
B
From (1),
`((z_(2)-z_(1))(z_(3) -z_(1)))/(z_(2) -z_(1)) =2((AD)/(IA))^(2)((AC)/(AD)) " " (because AB = 2AD)`
`rArr (z_(2) -z_(1)) (z_(3) -z_(1)) =(z_(4) -z_(1))^(2) 2 cos^(2). (theta)/(2) sec theta`
`= (z_(4) -z_(1))^(2) (cos theta + 1)`
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