Section 1.2 questions - using the definition?

[nadia.chigmaroff]

Hi,

For the first question on the Section 1.2 questions - $|z-4| = 4|z|$, what level of description is sufficient?
Can I say that this is a circle by the formula for Apollonius circles, and can I use the formula provided in lecture/the textbook to describe the radius and center point?
Should I describe the radius and center point, or is saying that it's a circle enough?

Thank you!

[RunboZhang]

Hi, below this comment I have attached my answer to the question. Basically we can use either of these two methods to solve for this kind of problem. So the answer for your first question is yes. For your second question, I think a general solution to this one should definitely include the locus of z, which is the function of the circle. It can be either in the form of

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(x-x_0)^2 +(y-y_0)^2 = R^2 or in the form of

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|z-z_0|=R.

[Victor Ivrii]

You need to derive it. Even the following argument would be insufficient for a full mark:

We know that it will be a circle, we know that its center will be on the real axis (left from $z=0$), so find two points of intersection of this circle with $x$-axis:
$(z_1-4)= -4z_1$ and $(z_2-4)= 4z_1$. Finding $z_{1,2}$ (even if you do it) and setting $z_0=(z_2+z_1)/2$, $R- |z_2-z_1|/2$ ...