If two triangles whose vertices are respectively the complex numbers `z_(1),z_(2),z_(3)` and `a_(1),a_(2),a_(3)` are similar, then the determinant.
`|{:(z_(1),a_(1),1),(z_(2),a_(2),1),(z_(3),a_(3),1):}|` is equal to

Options

A. `z_(1)z_(2)z_(3)`
B. `a_(1)a_(2)a_(3)`
C. 1
D. 0

Answer

D

Explanation

Since triangle ABC and DEF are similar.
`therefore angleCBA=angleEFD=theta`(say) and `(AB)/(BC) =(DE)/(EF)`.

Now, `(z_(1)-z_(2))/(z_(3)-z_(2))=(AB)/(BC)(costheta+isintheta)`
and `(a_(1)-a_(2))/(a_(3)-a_(2))-(DE)/(EF)(costheta+isintheta)`
` (z_(1)-z_(2))/(z_(3)-z_(2))=(a_(1)-a_(2))/(a_(3)-a_(2))`
`|{:(z_(1)-z_(2),a_(1)-a_(2)),(z_(3)-z_(2),a_(3)-a_(2)):}|=0 rArr |{:(z_(1)-z_(2),a_(1)-a_(2),0),(z_(3)-z_(2),a_(3)-a_(2),0),(z_(2),a_(2),1):}|=0`
`rArr |{:(z_(1),a_(1),1),(z_(2),a_(3),1),(z_(3),a_(2),1):}|=0`