We are given that |z-1| ≤ 2 and z₁ = 5 + 3i.


To solve this problem, we will use the geometric interpretation of complex numbers.


We are given that |z-1| ≤ 2, which represents a circle centered at 1 with a radius of 2. Let's call this region R.


We are also given z₁ = 5 + 3i, which represents a point in the complex plane. Let's call this point P.


To visualize the problem, let's plot the point P in the complex plane.

P = 5 + 3i

Plotting this point on the complex plane, we find that it lies in the first quadrant.


We need to find the maximum value of |iz z₁|.

Let's consider a point Q in region R such that Q lies on the line segment joining the origin O and P.

Let Q = tP, where t is a real number.

We need to find the maximum value of |iz z₁|, which is equal to |i(tP)z₁| = |itPz₁|.

Since |itPz₁| = |tiz₁P| and |tiz₁P| = |t| |iz₁P|, we can conclude that the maximum value of |iz z₁| will occur when t = 1.

Therefore, the maximum value of |iz z₁| is equal to |i(1)(5 + 3i)| = |i(5 + 3i)| = |5i - 3| = |3 - 5i|.

Using the distance formula, we can calculate |3 - 5i| as √((3 - 0)² + (-5 - 0)²) = √(9 + 25) = √34.

So, the maximum value of |iz z₁| is √34, which is approximately 5.83.

Therefore, the given answer of 7 is incorrect. The correct answer is approximately 5.83.


The maximum value of |iz z₁| is approximately 5.83, not 7. This is obtained by finding the point Q in region R that lies on the line segment joining the origin O and point P, and calculating the distance from the origin to Q.