Question

If complex number z lies on the curve `|z - (- 1+ i)| = 1`, then find the locus of the complex number `w =(z+i)/(1-i), i =sqrt-1`.

Answer 1

Correct Answer - A circle with centre at `(-3//2,1//2)` and radius `1//sqrt(2)`
We have`|z + 1 -i|= 1" "(1)`
and `w = (z +i)/(1-i)`
`therefore w ( 1- i) = z +i`
` rArr z = w (1-i) -i`
Putting the value of z in (1), we get
`|w (1-i) -i + 1 -i|=1`
`rArr | 1-i||w +(1-2i)/(1-i)|=1`
`rArr |w+(3-i)/(2)| = (1)/(sqrt(2))`
`rArr |w + (-3 +i)/(2)| = (1)/(sqrt(2))`
Therefore, locus of w is a c ircle havig centre at `(-3//2),1//2)` and radius `(1)/(sqrt(2))`.