Let's break down the questions visible in the image and solve them one by one.

Question 15:

You are asked to find the greatest and least values of $|z + 3 - 2i|$, given that $|z - (5 + 4i)| \leq 3$.

• This describes a region where the complex number $z$ lies inside or on the boundary of a circle with center $5 + 4i$ and radius 3.
• The expression $|z + 3 - 2i|$ measures the distance from $z$ to the point $-3 + 2i$.
• To solve this problem, we need to find the maximum and minimum distances between two points $5 + 4i$ and $-3 + 2i$ plus or minus 3 (the radius).

Solution Outline:

1. Circle Definition: The equation $|z - (5 + 4i)| \leq 3$ describes a circle with center $5 + 4i$ and radius 3.

2. Find Distance Between Centers: The distance between $-3 + 2i$ and $5 + 4i$ is given by $d = |(5 + 4i) - (-3 + 2i)| = |8 + 2i| = \sqrt{8^2 + 2^2} = \sqrt{64 + 4} = \sqrt{68} \approx 8.25.$

3. Greatest and Least Values:

   • Greatest Value: The maximum distance is $d + 3 = 8.25 + 3 = 11.25$.
   • Least Value: The minimum distance is $d - 3 = 8.25 - 3 = 5.25$.

   Therefore, the greatest and least values of $|z + 3 - 2i|$ are 11.25 and 5.25, respectively.

Question 16:

You are given that $|z - 3| = 2 |z - (3 + 9i)|$, and are asked to: (i) Show that the locus of $z$ is a circle, state its center and radius. (ii) Sketch the locus of the circle on an Argand diagram.

Part (i) Solution Outline:

1. Let $z = x + yi$, where $x$ and $y$ are real numbers representing the complex number $z$.

2. The given equation is $|z - 3| = 2 |z - (3 + 9i)|$. This implies a geometric relationship between distances: $|z - 3| = 2 |z - (3 + 9i)|,$ which means that the distance from $z$ to $3$ is twice the distance from $z$ to $3 + 9i$.

3. This defines a locus of points that satisfy this condition, which turns out to be a circle (based on the standard equation derived from the distances).

4. Using algebra, we solve this equation to find the equation of the circle. Expanding both sides gives us the standard form of a circle.

   The center and radius can then be computed algebraically. The circle will have a specific center and radius based on these computations.

Part (ii) Sketch:

• The circle can be sketched on an Argand diagram by plotting its center and radius, found from part (i).

Would you like a detailed breakdown of the algebra for Question 16 or any further clarifications?

Related Questions:

1. How do you compute the distance between two complex numbers geometrically?
2. Can the condition $|z - 3| = 2 |z - (3 + 9i)|$ be rewritten as a standard equation for a circle?
3. What is the geometric interpretation of the modulus of a complex number?
4. How can you find the equation of a circle given two points and a distance ratio?
5. How do you sketch a circle on an Argand diagram?

Tip:

When dealing with distances involving complex numbers, always remember that the modulus $|z - a|$ represents the Euclidean distance between the points $z$ and $a$ in the complex plane.