0
$\begingroup$

What is the image of $|z-4i|+|z+4i|=10$?

I tried to simplify this equation but it is too difficult for me. I tried squaring both side but it was too long and I can't get equation like any curve.

Can anyone give me any simple simplification? Or is there any result of complex number used in this example? Please help me. Thank in advance.

  • 1
    Similar questions: [Plot $|z-i|+|z+i|=16$ on the complex plane](http://math.stackexchange.com/questions/126518/plot-z-i-z-i-16-on-the-complex-plane) and [sketch set satisfying $|z-2|+|z+2|\le5$](http://math.stackexchange.com/questions/151972/sketch-set-satisfying-z-2z2-le5).2012-08-04

2 Answers 2

3

If we represent $Z$ by $x + iy$, then the equation above given by you after simplification becomes that of an ellipse: $25x^2 + 9y^2 = 225.$

Steps for simplification are:

  1. Insert the value of $Z$ as $x + iy$ and apply the magnitude formula of the complex numbers: $\sqrt{x^2 + y^2}$

  2. Take the part obtained from $|z+4i|$ to the RHS and then square both the sides; you will get on simplification

    $\sqrt{x^2 + (y-4)^2} + \sqrt{x^2 + (y+4)^2} = 10$

    $\sqrt{x^2 + (y-4)^2} = 10 - \sqrt{x^2 + (y+4)^2}$

    (square both sides)

    $x^2 + y^2 +16 -8y = 100 +x^2 + y^2 +16 +8y - 20\sqrt{x^2 + (y+4)^2}$

    (removing common terms and common factors)

    $4y + 25 = 5 \sqrt{x^2 + (y+4)^2}$

  3. Again square both sides and then simplify to obtain the equation of ellipse

    $16x^2 + 625 +200y = 25x^2 + 25 y^2 +200y + 400$

    $25x^2 + 9y^2 = 225$

  4. Process the output as you want to take the image.

  • 0
    great hints.thank you.2012-08-04
4

You might try imagining what the equation is saying in geometric terms. (Imagine complex numbers as points in the complex plane.)

To do that, notice that $|z-4i|$ is just the distance from $z$ to the point $4i$ in the complex plane. Similarly, $|z+4i|=|z-(-4i)|$ is the distance from $z$ to $-4i$.

The equation is saying that the sum of these two distances is equal to $10$ for all $z$ in your curve. But this is the familiar definition of the ellipse, quoting Wikipedia: an ellipse is "the set of points such that the sum of the distances to two fixed points (the foci) is constant". In our case the foci are $4i$ and $-4i$ and the constant is $C = 10$.

Note that from these numbers you may also calculate (using the usual formulas from elementary geometry, for example the ones given on Wikipedia*) the major ($a=5$) and minor ($b=3$) axes of the ellipse, so this gives you all the information needed to completely describe it.

(*): in our case, we know the distance from the center of the ellipse to each of the foci, it is given by $f=4$. We also know the constant sum of the distances from the foci, $C=10$. Now, simply use the formulas $f^2 = a^2 - b^2$ and $2a=C$ and the problem is solved.

  • 0
    @SiddhantTrivedi: From $2a=C=10$ we get $a=5$. From $f^2=a^2-b^2$ we get $4^2=5^2-b^2$ which is the same as $b^2=9$. Therefore $b=3$.2012-08-04