If P andn Q are represented by the complex numbers `z_(1)` and `z_(2)` such that `|1//z_(2)+1//z_(1)|=|1//z_(2)-1//z_(1)|`, then

Options

A. `DeltaOPQ` (where O is the origin) is equilateral.
B. `DeltaOPQ` is right angled
C. the circumcentre of `DeltaOPQ` is `(1)/(2)(z_(1) + z_(2))`
D. the circumcentre of `DeltaOPQ` is `(1)/(2)(z_(1)-z_(2))`

Answer

B, C

Explanation

We have
`|(1)/(Z_(2))+(10)/(z_(1))| = |(1)/(z_(2))-(1)/(z_(1))|`
`rArr |z_(1)+z_(2)| = |z_(1)-z_(2)|`
Squaring both side, we have
`|z_(1)|^(2) +|z_(2)|^(2) + 2(z_(1)barz_(2) + barz_(1)z_(2)) = 0`
`=|z_(1)|^(2)+|z_(2)|^(2) -2(z_(1)barz_(2) + barz_(1)z_(2))`
`rArr 4(z_(1)barz_(2) = barz_(1)z_(2)) = 0`
`rArr (z_(1))/(z_(2)) + (barz_(1))/(z_(2)) = 0`
`rArr arg ((z_(1))/(z_(2))) = (pi)/(2) = arg ((z_(1)-0)/(z_(2) -0))`

The angle between `z_(2)` O, and `z_(1)` is a right angle, taken in order lt brgt as shown in the above diagram ,. Now the circumcentre of the above diagram will lie on the PQ as diameter and is represented by C which is the centre of PQ, such that `z = (z_(1) + z_(2))//2` where z is the affix of circumceter.