Question

the locus of the complex number z such that Iz+3I = Iz-1I is?

A.     A line

B.     A circle

C.     A quadratic curve

An ellipse 

Answer 1

Given that $|z+3|=|z-1|$

Let $z=x+iy$

$|x+iy+3|=|x+iy-1|\\ \Rightarrow \sqrt{(x+3)^2+y^2}=\sqrt{(x-1)^2+y^2}\\$

Squaring both sides, we get:

$(x+3)^2+y^2=(x-1)^2+y^2\\ \Rightarrow x^2+9+6x=x^2+1-2x\\ \Rightarrow 8x+8=0\\ \Rightarrow x=-1$

Therefore, the locus of a given complex number is a line.

Hence, the correct option is (A).