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If $\dfrac{|2z-3|}{|z-i|}=k$ is the equation of circle with complex number $i$ lying inside the circle, find the values of $k$.

This post mentions center and radius of a general such circle. But I think that would be complicated, especially after looking at the book's solution. They have done $$\frac{|z-\frac32|}{|z-i|}=\frac k2\gt1\implies k\gt2$$

I didn't understand why it would be greater than $1$. Looks like the denominator $i$ has something to do with it. Can we do this question without finding center and radius?

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If $\frac{|z - 3/2|}{|z - i|} = w$, then $z$ is $w$ times as far away from $3/2$ as it is from $i$.

Suppose that $i$ is in the circle. Then trace a line from $3/2$ to $i$, and consider the two points $z_1, z_2$ where this line meets the circle. In particular, consider since $i$ is inside the circle, $i$ must be between $z_1$ and $z_2$ along the line. Take $z_2$ such that $i$ is between $z_2$ and $3/2$ on the line. Then we see that $z_2$ is closer to $i$ than to $3/2$. This means that $w = \frac{|z_2 - 3/2|}{|z_2 - i|} > 1$.

Conversely, suppose that $w > 1$. Then consider the points $z_1 = \frac{1}{1 + w} 3/2 + \frac{w}{1 + w} i$ and the point $z_2 = \frac{1}{1 - w} i - \frac{w}{1 - w} 3/2$. Both of these points are on the line connecting $3/2$ and $i$, and both of these satisfy $\frac{|z - 3/2|}{|z - i|} = w$. So these are two points on the circle, and it's easy to see that $i$ lies between the two. Then $i$ is in the circle.

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  • $\begingroup$ Thankyou for the answer. $\endgroup$
    – aarbee
    Jul 15 '21 at 21:36
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Let's assume that you know that the locus is a circle. We write this as $$|z-z_{1}|=k|z-z_{2}|$$ So the circle is given by two points $z_1$ and $z_2$, and a ratio of the distances to these points. Say that the ratio $k$ is exactly equal to $1$. Then the locus is a straight line perpendicular to the line connecting the two points, exactly in the middle. For any other $k$ only one of $z_1$ or $z_2$ is inside of the circle.if $k<1$, then $z_1$ is inside, if $k>1$ then $z_2$ is inside. You know that a circle intersects a line in at most two points. Calculate the intersection of the circle for a given $k$ to see that my statement above is true.

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  • $\begingroup$ Thankyou for the answer. $\endgroup$
    – aarbee
    Jul 15 '21 at 21:36

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