Question

If `z=x+i y ,` then show that `z bar z +2(z+ bar z )+a=0` , where `a in R ,` represents a circle.

Answer 1

`Z=x+iy`
`Z Z=|Z|^2`
`|sqrt(x^2+y^2)|=x^2+y^2`
`|Z|^2+2(Z+Z)+a=0`
`x^2+y^2+2(x+iy+x-iy)+a=0`
`x^2+y^2+4x+a=0`
`x^2+4+4x+y^2+a-4=0`
`(x+2)^2+y^2=(sqrt(4-a))^2`
`(x-h)^2+(y-k)^2=r^2`
`(h,k)` Centre
(-2,0),`r=sqrt(4-a)`.