Question

What is the value of z/4 if z = 20

Answer 1

We have for any two complex numbers α and β

∣∣α∣∣≤∣α−β∣

Now ∣∣Z∣−∣

∣Z∣

4

∣∣≤∣Z−

Z

4

∣

⇒∣Z∣−

∣Z∣

4

∣≤2

Set ∣Z∣=r>0, then ∣r−

r

4

∣≤2

⇒−2≤r−

r

4

≤2

The left inequality gives

r

2

+2r−4≥0

The corresponding roots are

r=

2

−2±

20

=−1±

5

Thus r≥

5

−1 or r≤−1−

5

implies that r≥

5

−1 (As r > 0) ... (i)

Again consider the right inequality

r−

r

4

≤2⇒r

2

−2r−4≤0

The corresponding roots are

r=

2

2±

20

=1±

5

Thus 1−

5

≤r≤1+

5

But r > 0, hence r≤1+

5

.... (ii)

(i) and (ii) gives

5

−1≤r≤

5

+1

So, the greatest value is

5

+1.

Answer 2

part 1. part 2

hope it helps you