If `z^2+z|z|+|z^2|=0,` then the locus `z` is a. a circle b. a straight line c. a pair of straight line d. none of these

Options

A. a circle
B. a straight line
C. a pair of straing line
D. none of these

Answer

C

Explanation

`z^(2)+z|z|+|z|^(2)=0`
`rArr ((z)/(|z|))^(2)+(z)/(|z|)+1=0`
`rArr (z)/(|z|)=omega,omega^(2)`
`rArr z-omega |z|orz=omega^(2)|z|`
`rArr x+iy=|z|((-1)/(2)+(isqrt3)/(2))orx+iy=|z|((-1)/(2)-(isqrt3)/(2))`
`rArr =-(1)/(2)|z|,y=|z|(sqrt3)/(2)or x=-(|z|)/(2),y=(|z|sqrt3)/(2)`
`rArr y+sqrt3x=0ory-sqrt(3)x=0rArr y^(2)-3x^(2)=0`