## SheldonEinstein 3 years ago Let z be a complex number than locus represented by |iz-1| + |z-i| = 2 is : a) a line , b) a circle , c) a pair of straight lines, d) a parabola

1. satellite73

i think it is a l ine

2. SheldonEinstein

Sorry for the mistake in the question earlier , the modified quest. is in the post itself. I seek help from the users presented here as soon as possible,

3. SheldonEinstein

4. satellite73

scratch that, i think it is a region bounded by two line

5. SheldonEinstein

OK so since it is IIT based so I will say : There may be no answers or multiple answers

6. SheldonEinstein

Well yes @satellite73 has the answer, it will be a line $$\textbf{segment}$$

7. SheldonEinstein

But I am half way stuck @satellite73 , can you show your work, I think @experimentX is also writing his work :)

8. experimentX

put z = x + iy you would get, $\sqrt{(y-1)^2 + x^2} + \sqrt{x^2 + (y-1)^2} = 2$ http://www.wolframalpha.com/input/?i=plot+ \sqrt{%28y-1%29^2+%2B+x^2}+%2B+\sqrt{x^2+%2B+%28y-1%29^2}+%3D+2 guess it is a circle

9. experimentX

woops!! (y+1)^2

10. experimentX
11. SheldonEinstein

What I can do is : $\large{|(-iz-1)| + |(z-i)| =2}$ $\large{|i(z+i)|+|z-i|=2}$ $\large{|i||z+i| + |z-i| = 2 }$ $\large{|z+i| + |z-i| = 2}$

12. satellite73

that looks good

13. SheldonEinstein

|dw:1352908224557:dw|

14. SheldonEinstein

So am I right in my way @experimentX @satellite73 ? (Since I am only given 54 seconds to do these type of questions so I prefer to use shortcuts that is here : not putting z = x + iy , my opinion said , yes ! x will be equal to zero if we are going to solve it! )

15. satellite73

yes $$x=0$$

16. satellite73

i did it the donkey way but your way is much much better i think

17. UnkleRhaukus

$$z=i$$ is a solution

18. SheldonEinstein

Oh! No problem, sometimes we learn a lot from "that say donkey ways" , thanks for your time : @experimentX @satellite73

19. experimentX

|z| <= 1 and z = i

20. experimentX

no wonder it does not show in x's and y's

21. UnkleRhaukus

$$z=-i$$ is also a solution

22. experimentX

the solution is a line segment from +i to -i

23. UnkleRhaukus

$\large{|z+i| + |z-i| = 2}$ change of variables shift along the complex plane $w=z+i$ ${|w| + |w-2i| = 2}$

24. UnkleRhaukus

|dw:1352909335544:dw|

25. UnkleRhaukus

circle + circle = circle?