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

Question

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.

Similar questions: Plot $|z-i|+|z+i|=16$ on the complex plane and sketch set satisfying $|z-2|+|z+2|\le5$. — Martin Sleziak, Aug 04 '12 at 12:17

score 4 · Accepted Answer · edited Aug 04 '12 at 13:34

4

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:

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

$4y + 25 = 5 \sqrt{x^2 + (y+4)^2}$
Again square both sides and then simplify to obtain the equation of ellipse

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

$25x^2 + 9y^2 = 225$
Process the output as you want to take the image.

edited Aug 04 '12 at 13:34

Vincenzo Tibullo

10,955

answered Aug 04 '12 at 11:52

nimeshkiranverma

614

can you write simplification in your answer? – Siddhant Trivedi Aug 04 '12 at 12:00
I guess it is pretty much clear now. This a standard IIT-JEE question which you can try with the equation of circles in complex form – nimeshkiranverma Aug 04 '12 at 12:12
great hints.thank you. – Siddhant Trivedi Aug 04 '12 at 12:16

Dejan Govc · Answer 2 · 2012-08-04T12:25:15.057

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.

you r right. geometrically it was easy to see that but i want some easy way to simplify this equation. — Siddhant Trivedi, Aug 04 '12 at 11:58
@SiddhantTrivedi: I have added the relevant details of the calculation. I hope this helps. — Dejan Govc, Aug 04 '12 at 12:27
please explain me.i want to know how u find a and b with help of constant and focus? — Siddhant Trivedi, Aug 04 '12 at 13:12
@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 $b=3$. — Dejan Govc, Aug 04 '12 at 15:34

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

2 Answers2