Complex number question (1 Viewer)

[Kipling]

The equation |z-1-3i| + |z-9-3i| = 10 corresponds to an ellipse in the Argand diagram.

It asks to write down the complex number corresponding to the centre of the ellipse. When I see these types of questions I always sub in x+iy for z and do the whole square rule thing, however is there a better way to do this? I don't think this is a hard question but for some reason just can't do it

 

[Sy123]

The equation |z-1-3i| + |z-9-3i| = 10 corresponds to an ellipse in the Argand diagram.

It asks to write down the complex number corresponding to the centre of the ellipse. When I see these types of questions I always sub in x+iy for z and do the whole square rule thing, however is there a better way to do this? I don't think this is a hard question but for some reason just can't do it

The centre of the ellipse is middle of the 2 focii, we know that the focii is (1,3) and (9,3), hence the centre is (5,3)

First geometrically interpret the question, z is some number in the complex plane so that its distance from (1,3) (given by |z-1-3i|) plus the distance to (9,3) (given by |z-9-3i|) is constant at 10.
Recognise that this is an alternate definition for the ellipse.
That is, an ellipse is the locus where the sum of distances from variable point to 2 focii is constant. Therefore those 2 co-ordinates are focii.