Here's the question you clicked on:

henpen Group Title A complex question 2 years ago 2 years ago

• This Question is Closed
1. henpen

$\left| z-1 \right|+\left| z+1 \right|=4$ 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. henpen

(Where z=a+ib)

3. henpen

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

4. henpen

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. baddinlol

OK

6. baddinlol

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

7. baddinlol

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

8. baddinlol

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

9. baddinlol

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

10. baddinlol

Then you should square both sides

11. baddinlol

Actually that will be hard

12. baddinlol

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

13. baddinlol

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

14. baddinlol

Which will give you this

15. henpen

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

16. baddinlol

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

17. baddinlol

I will get rid of it soon

18. baddinlol

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

19. baddinlol

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

20. baddinlol

Now i can simplify further

21. baddinlol

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

22. henpen

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

23. baddinlol

Now I am going to square both sides again

24. baddinlol

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

25. baddinlol

Finally we have no more square roots

26. baddinlol

Now i am going to expand

27. baddinlol

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

28. henpen

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

29. baddinlol

Now simlify...

30. baddinlol

SSo you get

31. henpen

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. baddinlol

Final step: complete the square on a term

33. baddinlol

oops

34. baddinlol

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

35. baddinlol

And thats an eclipse.

36. baddinlol

No problem. =)