![]() Since we don't need to find the exact points, we can more easily find the distance between $-13+5i$ and $-1$, the center of the circle that represents $iz$. ![]() To find the maximum, you extend the line segment until you find another crossing out of the disc, that would be the point of maximum distance. To find the minimum, you then go back along that line segment to the point where the segement crosses into the disc, that's your point of smallest distance. That's easy, you connect the fixed point to the center of the disc. So what's the minimum/maximum distance from a fixed point to a disc? $-13+5i$ is a fixed point, and as we saw above $iz$ is a disc of radius $4$ around the point $-1$. That's the distance from the point $iz$ to the point $-13+5i$. That means the set if points $iz$ is the disc around $-1$ with radius 4. To find the center of the rotated disc, we just multiply the center of the original disc with the factor: $i\times i=-1$, so the rotated disc is around the center point $-1$, and of course it still has radius $4$. In this case, as the factor is $i$, the absolute value is $1$, the whole operation is just a rotation. Multiplication by a fixed complex number means a rotation around the origin, with the argument of the factor as rotation angle, and then a stretching from the origin, with the absolute value of the factor as stretch factor. If you now consider the value $iz$, that's $z$ multiplied by $i$. ![]() That means it contains exactly the complex values $z$ which have a distance of $4$ or less from $i$. One way to solve both the maximum and the minimum problem is to consider what the various inequalities mean in the complex plane:ĭescribes a disk of radius $4$ around $i$.
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