Question

`z_1a n dz_2` are two distinct points in an Argand plane. If `a|z_1|=b|z_2|(w h e r ea ,b in R),` then the point `(a z_1//b z_2)+(b z_2//a z_1)` is a point on the line segment [-2, 2] of the real axis line segment [-2, 2] of the imaginary axis unit circle `|z|=1` the line with `a rgz=tan^(-1)2`
A. line segment `[-2,2]` of the real axis
B. line segment `[-2,2]` of the imaginary axis
C. unit circle `|z|=1`
D. the line with arg `z = tan ^(-1)2`

Answer 1

Correct Answer - A
Assuming arg `z_(1)=theta` and `arg z_(2)theta+alpha`,
`(az_(1))/(bz_(2))+(bz_(2))/(az_(1))=(a|z_(1)|e^(itheta))/(b|z_(2)|e^(i(theta+alpha)))+(b|z_(2)|e^(i(theta+alpha)))/(a|z_(1)|e^(itheta))`
Hence, the point lies on the line segment `[-2,2]` of the real axis.