Question

If z−3+2i ≤ 4 then the differencebetween the greatest value and the leastvalue of z is

Answer 1

Answer:

4 + √13

Step-by-step explanation:

Hi,

|z-z₀| ≤ c represents the entire circular region, not just the curve of circle(its

entire area is included) , where center of circle is z₀ and radius of circle is c.

|z-z₀| = c  just represents the circle(i.e the circumference alone)

Note:

greatest value and the least value of z doesn't make any sense , since

Order is not defined for complex numbers(i.e., we cannot compare

compare complex numbers)

So, it should be modulus value of z , |z|(distance from origin).

Maximum value of |z| = c + √(a² + b²) , when z is situated at the farthest

point from origin along the line joining origin and center.

When it comes to minimum value , since entire circular region is included,

even z could be at origin itself and hence minimum value of |z| would be 0.

here, |z - 3 + 2i| is similar to |z - z₀| . z₀ = 3 - 2i

and c is 4.

now, if |Z - 3 + 2i| ≤ 4 ,

maximum value of |z| = 4 + √13

and minimum value of |z| = 0

hence, difference of maximum to minimum = (4 + √13) - 0 = 4 + √13

Hope, it helped !