## anonymous 3 years ago A complex question

1. anonymous

$\left| z-1 \right|+\left| z+1 \right|=4$ Which turns into the messy $\sqrt{(a-1)^2+b^2}+\sqrt{(a+1)^2+b^2}=4$ Which I have no idea about how to solve graphically.

2. anonymous

(Where z=a+ib)

3. anonymous

(And by 'graphically' I mean plotting the possible values of z on the complex plane)

4. anonymous

http://www.wolframalpha.com/input/?i=sqrt%28%28x%2B1%29%5E2%2By%5E2%29%2Bsqrt%28%28x-1%29%5E2%2By%5E2%29%3D4 Can anyone explain why the graph is an ellipse?

5. anonymous

OK

6. anonymous

| a + bi - 1| + |a + bi + 1| = 4

7. anonymous

| (a -1) + (b)i | + | (a+1) + (b)i | = 4

8. anonymous

Now let me find the modulus which is given by sqrt( re(z)^2 + Im(z)^2)

9. anonymous

$\sqrt{(a +1)^2 + b^2} + \sqrt{(a-1)^2 + b^2} = 4$

10. anonymous

Then you should square both sides

11. anonymous

Actually that will be hard

12. anonymous

So move one of the square roots on the left then square both sides

13. anonymous

$(\sqrt{(a+1)^2 + b^2} )^2= (4 - \sqrt{(a-1)^2 + b^2})^2$

14. anonymous

Which will give you this

15. anonymous

There's still a horrible square root sign there, though.

16. anonymous

$(a+1)^2 + b^2= 16 - 8\sqrt{(a-1)^2 + b^2} + (a-1)^2 + b^2$

17. anonymous

I will get rid of it soon

18. anonymous

Ok now i am going to cancel some stuff and simplify this equation

19. anonymous

$a^2 + 2a + 1 + b^2 = 16 - 8\sqrt{(a-1)^2 + b^2} + a^2 - 2a + 1 + b^2$

20. anonymous

Now i can simplify further

21. anonymous

$4a -16 = -8\sqrt{(a-1)^2+b^2}$

22. anonymous

4a-16=-8sqrt(a^2-2a+1+b^2)

23. anonymous

Now I am going to square both sides again

24. anonymous

$(4a -16)^2 = 64((a-1)^2 + b^2)$

25. anonymous

Finally we have no more square roots

26. anonymous

Now i am going to expand

27. anonymous

$16a^2 -128a + 256 = 64 a^2 -128a + 64 + 64b^2)$

28. anonymous

it ends up as 192=48a^2+64b^2 I think

29. anonymous

Now simlify...

30. anonymous

SSo you get

31. anonymous

That actually agrees with the wolfram link I posted earlier. Thanks very much- I had little idea how to deal with square roots efficiently before!

32. anonymous

Final step: complete the square on a term

33. anonymous

oops

34. anonymous

Sorry yeah that was the answer 192 = 48a^2 +64b^2

35. anonymous

And thats an eclipse.

36. anonymous

No problem. =)