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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 `(z_(4) -z_(1))^(2) (cos theta + 1) 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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