complex numbers

[cazza90]

Find the Cartesian equation for the region represented by
| z-6+3 i | = 1/5*| z+4+2 i |

Please put your answer in a "natural" form.

So i substituted z= x + iy in and then used the |U + iv|^2 = u^2 + v^2 formula - now what am i supposed to do? what is natural form??

 

[mmm4444bot]

I can't see your work, but I think the following form is a "natural" choice.

\(\displaystyle \frac{4}{5} \cdot x^2 \;-\; \frac{4}{5} \cdot y^2 \;-\; \frac{68}{5} \cdot x \;-\; \frac{26}{5} \cdot y \;+\; \frac{123}{5} \;=\; 0\)

If you prefer Integers, try Standard Form:

4x^2 - 4y^2 - 68x - 26y + 123 = 0

or General Form:

4x^2 - 4y^2 - 68x - 26y = -123

I hope that I interpreted the original exercise correctly.

Cheers ~ Mark

 

[soroban]

Hello, cazza90!

\(\displaystyle \text{Find the Cartesian equation for the region represented by: }\;|z\,-\,6\,+\,3i| \:=\: \tfrac{1}{5}| z\,+\,4\,+\,2i|\)
Please put your answer in a "natural" form.

\(\displaystyle \text{Rewrite the equation: }\;|z - (-4-2i)| \;=\;5|z - (6-3i)|\)

\(\displaystyle \text{The left side is the distance frim }z\text{ to point }A:\;-4-2i \:=\-4,-2)\)

\(\displaystyle \text{The right side is 5 times the diatance from }z\text{ to pont }B:\;6-3i \:=\6,-3)\)


\(\displaystyle \text{We have: }\;\sqrt{(x+4)^2 + (y+2)^2} \;=\;5\sqrt{(x-6)^2 + (y+3)^2}\)


\(\displaystyle \text{Expanded to "natural" form: }\;24x^2 + 24y^2 - 308x + 146y +1105 \;=\;0\)

. . \(\displaystyle \text{(It is called the Circle of Apollonius.)}\)

Corrected my typo . . .

 

[mmm4444bot]

\(\displaystyle 24x^2 + 24y^2 - 308x + 146y - 1105 = 0\)

Is the -1105 a typographical error ?

I found my mistake, but now I have:

24x^2 + 24y^2 - 308x + 146y + 1105 = 0