To determine the radius r of the circle, let's analyze the geometric relationships in the Argand diagram.

First, consider the points where the line L intersects the real and imaginary axes. These points are 20 on the real axis and 15\mathrm{i} on the imaginary axis.

The distance between these two points can be calculated using the Pythagorean theorem:

\text{Length of } L = \sqrt{20^2 + 15^2} = \sqrt{400 + 225} = \sqrt{625} = 25

Next, observe that the circle is tangent to the line L as well as both the real and imaginary axes. This means that the circle touches the real axis and the imaginary axis at exactly one point each.

Let’s denote the points of tangency on the real and imaginary axes as A and B , respectively. Since the circle is tangent to both axes, the distances from the points 20 and 15i to these points of tangency are reduced by the radius r :

The distance from 15i to the point of tangency B on the imaginary axis is 15 - r .
The distance from 20 to the point of tangency A on the real axis is 20 - r .

Because the line L is tangent to the circle, and the length of two tangents to a circle are always the same, the total length of L is equal to the sum of these two reduced distances:

25=(15-r)+(20-r)

Simplifying the equation:

25 = 35 - 2r

2r = 10

r = \boxed{5}

General

Complex Numbers - Practice problems