locus of complex $z=(λ+3) + i\sqrt{3-λ^2}$

Question

if $z=(λ+3) + i\sqrt{3-λ^2}$, for all real $λ$, then the locus of $z$ is ? Please help.

Options are

• (A) circle
• (B) parabola
• (C) line
• (D) none of these

Answer 1

Why don't you try to evaluate the square of the modulus? This sometimes helps a lot: $$|z|^2=\lambda^2+6\lambda+9+3-\lambda^2=6\lambda+12$$

From here (or directly: upon taking the modulus $\;\lambda^2\;$ gets "killed" , and thus we only need to "kill" that $\;6\lambda\;$) we can see that

$$|z-3|^2=3$$

and this is a circle. This is the same JJaquelin got with more "real-plane" methods.

Answer 2

$x=\lambda+3$

$y^2=3-\lambda^2$

$y^2=3-(x-3)^2$

$y^2+(x-3)^2=3$

Do you know what kind of curve it is ?