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

try to draw this in Matlab and see what it looks like. You can also consider $x=\lambda+3$ and $y=\sqrt{3-\lambda^2}$ and see what the curve looks like — enigne, May 13 '14 at 08:21

score 2 · Answer 1 · answered May 13 '14 at 08:51

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 ?

answered May 13 '14 at 08:51

JJacquelin

score 0 · Accepted Answer · answered May 13 '14 at 10:34

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.

2 Answers2