**Re: HSC 2018 MX2 Marathon**

i)

i.e.

. But,

Therefore the locus of

is the line

, which is linear and passes through

.

The exception is when

, where the locus of

is all complex numbers

. Going off the question, I will assume

is nonzero.

ii)

Using a substitution obtainable from the locus from i),

, we have

Simplify and we get the quadratic equation

Let the roots be

. Then, using the product of roots formula,

Which I will call equation

. Now,

Using equation

,

Therefore,

, for

.

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Considering the case where

:

So, either of

or

or both are zero.

If

and it can be proved in a similar manner that

.

If

, which I addressed in part i).