If `|z-(4 + 3i) |=1,` then find the complex number z for each of the following cases:
(i) `|z| ` is least
(ii) `|z|` is greatest
(iii) `arg(z)` is least
(iv) arg(z) is greatest

Explanation

Given that `|z - (4+3i)|=1`
So, z lies on the circle having center C at 4+3i and radius 1.
Draw a line through origin and centre of the circle which meets circle at A and B .
Clearly complex number for point A has the least moduls and that for point B has the greatest modulus.

`therefore OA = OC - AC = 5-1= " "("least value of |z|")`
and `OB = OC + BC = 5+1=6" "(" greatest value of |z|")`
Also `1=|z-(4+3i)| ge ||z|-4 |4+3i||`
`therefore ||z|-|4+3i| | le 1`
`rArr -1 le |z| - 5 le 1`
`rArr 4 le|z| le 6`
Therefore , least value of |z| is 4 and greatest value is 6.
Now , `tan theta = (3)/(4)`, where `theta` is argument of complex numbers for point A and B.
For point A, `z = OA (cos theta + isin theta) = 4 ((4)/(5)+i(3)/(5)) = (16+12i)/(5)`
For point `B, z = OB (cos theta +isin theta) = 6((4)/(5)+i(3)/(5)) = (24+18i)/(5)`
Now, `OD = sqrt(OC^(2) - CD^(2)) = sqrt(25-1) = sqrt(24)`
Also `tan (theta -alpha) = (1)/(sqrt(24))`
`rArr ((3)/(4)-tan alpha)/(1+(3)/(4) tan alpha )= (1)/(sqrt(24))`
`rArr (6sqrt(6)-4)/(3+8sqrt(3))=tan alpha `
So, for point `D, z = sqrt(24) (cos alpha + isin alpha)`
Now, `tan (beta -theta ) = (1)/(sqrt(24))`
`therefore beta = (6sqrt(6) +4)/( 8sqrt(6)-3)`
So , for point `E, z = sqrt(24)(cos beta + isin beta)`