A complex number z lies on or within a circle centered at (2, 0) with radius 5 units. What is the maximum value of |z – 2|?

Solution

Let the complex number be z=x+iy. The circle centered at (2,0) in the complex plane (which corresponds to the complex number 2+0i=2) with radius 5 units can be described by the inequality: ∣ z−2 ∣ ≤5 Here, ∣ z−2 ∣ represents the distance between the complex number z and the center of the circle, which is 2. The condition ∣ z−2 ∣ ≤5 means that the point z lies on or within this circle. We want to find the maximum value of ∣ z−2 ∣ . The expression ∣ z−2 ∣ represents the distance between the point z and the point 2 (the center of the circle). Since z lies on or within the circle, the maximum distance between z and the center of the circle occurs when z is on the boundary of the circle and is farthest from the point 2. The distance from the center of the circle to any point on the circle is equal to the radius, which is 5. The expression ∣ z−2 ∣ is precisely this distance. The condition ∣ z−2 ∣ ≤5 directly tells us that the distance between z and 2 is at most 5. The maximum value of ∣ z−2 ∣ occurs when ∣ z−2 ∣ =5. This happens for any complex number z lying on the circle ∣ z−2 ∣ =5. Let's consider a geometric interpretation. The circle has its center at the point 2 on the real axis and a radius of 5. Any point z on or inside this circle satisfies the condition. The quantity ∣ z−2 ∣ is the distance from z to the center 2. The maximum value of this distance for points on or inside the circle is the radius of the circle. Maximum value of ∣ z−2 ∣ =5.