Pretty dumb question involving complex numbers

[pylauzier]

Homework Statement

I'm asked to describe geometrically the set of points in the complex plane describing some equations. I got them all right except this one:

|z+1| + |z-1| = 8

Homework Equations

|z| = sqrt( x² + y² )

The Attempt at a Solution

Well, I know that an equation of the |z+1| = 8 type would be a circle centered at (-1,0) with a radius of 8. Such a circle has the following equation:

(x+1)² + y² = 8²

I started from there to write my equation for |z+1| + |z-1| = 8 using x's and y's.


(x+1)² + y² + (x-1)² + y² = 8²

x² - 2x + 1 + x² +2x +1 + 2y² = 8²

2x² + 2y² = 14


I end up with the equation of a circle, but the solution manual says the solution is an ellipse
of foci (-1,0), (1,0), semi-major axis = 4. What did I do wrong?

 

[SammyS]

Homework Statement

I'm asked to describe geometrically the set of points in the complex plane describing some equations. I got them all right except this one:

|z+1| + |z-1| = 8

Homework Equations

|z| = sqrt( x² + y² )

The Attempt at a Solution

Well, I know that an equation of the |z+1| = 8 type would be a circle centered at (-1,0) with a radius of 8. Such a circle has the following equation:

(x+1)² + y² = 8²

I started from there to write my equation for |z+1| + |z-1| = 8 using x's and y's.


(x+1)² + y² + (x-1)² + y² = 8²

x² - 2x + 1 + x² +2x +1 + 2y² = 8²

2x² + 2y² = 14


I end up with the equation of a circle, but the solution manual says the solution is an ellipse
of foci (-1,0), (1,0), semi-major axis = 4. What did I do wrong?

Start with the basics.

\left|z\right|=\sqrt{z\bar{z}}=\sqrt{(x+iy)(x-iy)}=\sqrt{x^2+y^2}

|z+1|=\sqrt{(x+1)^2+y^2}

|z-1|=\underline{\ \ ?\ \ }

 

[pylauzier]

|z-1| = sqrt ((x-1)² + y² ), no?

 

[NewtonianAlch]

Well essentially you have two points given to you that are the foci of the ellipse.

What is the equation telling you in terms of length and the two points?

 

[pylauzier]

Well when I did the problem I didn't intuitively notice that the equation would give me an ellipse, that's why I developped it the way I did. I'm trying to figure out what I did wrong, since I got the equation of a circle.

Why isn't |z+1| + |z-1| = 8

equal to

sqrt [(x+1)² + y²] + sqrt[(x-1)² + y²] = 8

(x+1)² + y² + (x-1)² + y² = 8²

??

 

[cng99]

Exactly! You forgot the square root. That's what you did wrong.
If you notice, putting in the square root makes it look like the locus of points having the sum of distances from two fixed points equal to constant.
And that is what the definition of an ellipse is!

 

[pylauzier]

Exactly! You forgot the square root.

I didn't forget it, I simply started at the step right after it in my initial post. Notice the squared 8, I squared the equation to remove the square roots. If you simplify the (x+1)² and (x-1)² terms, you end up with 2x² + 2, so the equation becomes

2x² + 2y² = 8² - 2

x² + y² = 7

 

[SammyS]

Well when I did the problem I didn't intuitively notice that the equation would give me an ellipse, that's why I developped it the way I did. I'm trying to figure out what I did wrong, since I got the equation of a circle.

Why isn't |z+1| + |z-1| = 8

equal to

sqrt [(x+1)² + y²] + sqrt[(x-1)² + y²] = 8

The next step is incorrect.

(x+1)² + y² + (x-1)² + y² = 8²

??

(a+b)² ≠ a + b !

 

[pylauzier]

Oh god I'm an idiot, thanks everyone. I guess I shouldn't be doing maths at 1:30am when sleep deprived :P