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Complex numbers: loci on the Argand diagram

mathematical

The complex number

z = x + iy

satisfies

|z - 3| = 2|z + i|

The circle |z − 3| = 2|z + i| with centre (−1, −4/3); the straight line through the origin and the centre meets the circle at the points of greatest and least modulus.

Facts you may need

$|x + iy| = \sqrt{x^2 + y^2}$ — Square both sides to remove roots.
Completing the square: $x^2 + 2x = (x + 1)^2 - 1$ — To find the centre and radius.
Furthest point = distance to centre + radius; nearest = |distance − radius| — Along the line through the centre.

(a)

Show that the locus of

z

is a circle, and state its centre and exact radius.

(b)

Find the exact maximum value of

|z|

for points on this locus.

(c)

Find the exact minimum value of

|z|

, justifying your method.