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

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?

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.
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.