Question

Let `z_1, z_2, z_3` be three complex numbers and `a ,b ,c` be real numbers not all zero, such that `a+b+c=0a n da z_1+b z_2+c z_3=0.` Show that `z_1, z_2,z_3` are collinear.

Answer 1

Given
`a+ b+ c = 0" "(1)`
and ` az_(1) + bz_(2) + cz_(3) = 0" "(2)`
Since a, b c are not all zero, from (2), we have
`az_(1) bz_(2) -(a+b) z_(3) = 0 " "["From"(1), c = - (a+b)]`
or ` az_(1) + bz_(2) = (a+b)z_(3)`
or `z_(3) =(az_(1) + bz_(2))/(a+b) " "(3)`
From (3), it follows tha `z_(3)` divides the line segment joining `z_(1)` and `z_(2)` internally in the ratio b:a
Hence `z_(1), Z_(2) and z_(3)` are coliner.
If a and b are of same sign , then division is in fact internal,
and if a and b are of opposte sign, then division is external in the ratio `|b|:|a|`