Question

A complex question

Answer 1

\[\left| z-1 \right|+\left| z+1 \right|=4\]
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.

Answer 2

(Where z=a+ib)

Answer 3

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

Answer 4

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?

Answer 5

OK

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

Then you should square both sides

Answer 11

Actually that will be hard

Answer 12

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

Answer 13

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

Answer 14

Which will give you this

Answer 15

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

Answer 16

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

Answer 17

I will get rid of it soon

Answer 18

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

Answer 19

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

Answer 20

Now i can simplify further

Answer 21

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

Answer 22

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

Answer 23

Now I am going to square both sides again

Answer 24

\[(4a -16)^2 = 64((a-1)^2 + b^2)\]

Answer 25

Finally we have no more square roots

Answer 26

Now i am going to expand

Answer 27

\[16a^2 -128a + 256 = 64 a^2 -128a + 64 + 64b^2)\]

Answer 28

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

Answer 29

Now simlify...

Answer 30

SSo you get

Answer 31

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!

Answer 32

Final step: complete the square on a term

Answer 33

oops

Answer 34

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

Answer 35

And thats an eclipse.

Answer 36

No problem. =)