# Thread: complex no. + locus again

1. ## complex no. + locus again

p294 q16 c
given
$z^2-2z+k=0$
.......(*), where k is real, has no real roots.
find , in terms of k, the squares of the roots of (*), expressing the answers in the form x+iy where x and real are real.
as k varies, find the equation of the locus of the pts in the argand plane representing the squares of the roots of (*)

my working:
$z = \frac {2\pm \sqrt{4-4k}}2 = 1 \pm \sqrt{1-k}$
square of the roots $= 2-k \pm 2\sqrt{k-1}i$
don't know how to find the locus. thanks.

2. Originally Posted by afeasfaerw23231233
p294 q16 c
given
$z^2-2z+k=0$
.......(*), where k is real, has no real roots.
find , in terms of k, the squares of the roots of (*), expressing the answers in the form x+iy where x and real are real.
as k varies, find the equation of the locus of the pts in the argand plane representing the squares of the roots of (*)

my working:
$z = \frac {2\pm \sqrt{4-4k}}2 = 1 \pm \sqrt{1-k}$
square of the roots $= 2-k \pm 2\sqrt{k-1}i$
don't know how to find the locus. thanks.
It's certainly a pleasure to see some working and a clear statement of where you're stuck.

Note that since the roots are not real, k > 1.

Real part: $x = 2 - k$, k > 1 .... (1)

Imaginary part: $y = \pm 2 \sqrt{k - 1}$, k > 1 .... (2)

From (1): $k = 2 - x$ .... (3)

Substitute (3) into (2) to get the Cartesian equation of the locus (it's a sideways parabola).