Question

If `z_1, z_2, z_3` are three complex numbers such that `5z_1-13 z_2+8z_3=0,` then prove that `|z_1( z )_1 1z_2( z )_2 1z_3( z )_3 1|=0`

Answer 1

`5z_(1)-13z_(2)+8z_(3)=0`
or `(5z_(1)+8z_(2))/(5+8)=z_(3)`
This mean that `z_(3)` divides segment joining `z_(1)" and "z_(2)` in the ratio 8 : 5.
Hence, `z_(1),z_(2),z_(3)` are collinear.
`implies|{:(z_(1),bar(z)_(1),1),(z_(2),bar(z)_(2),1),(z_(3),bar(z)_(3),1):}|=0" "("condition of collinear points")`