Question

Find the relation between x and y of a complex number variable z=x+iy satisfying the condition

|
−3
/+3
| = 2.​

Answer 1

$\textbf{Given:}$

$\text{ z=x+i\,y is the variable complex number and |\dfrac{z-3}{z+3}|=2 }$

$\textbf{To find:}$

$\text{The relation between x and y}$

$\textbf{Solution:}$

$\text{Consider,}$

$|\dfrac{z-3}{z+3}|=2$

$\dfrac{|z-3|}{|z+3|}=2$

$|z-3|=2\,|z+3|$

$|(x+iy)-3|=2\,|(x+iy)+3|$

$|(x-3)+iy|=2\,|(x+3)+iy|$

$\sqrt{(x-3)^2+y^2}=2\,\sqrt{(x+3)^2+y^2}|$

$\text{Squaring on bothsides, we get}$

$(x-3)^2+y^2=4[(x+3)^2+y^2]$

$x^2+9-6x+y^2=4[x^2+9+6x+y^2]$

$x^2+9-6x+y^2=4x^2+36+24x+4y^2$

$\text{Rearranging terms, we get}$

$\boxed{\bf\,3x^2+3y^2+30x+27=0}\;\text{whihch is the required reation between x and y}$

Find more:

Solve the following equations in complex numbers and write your answer in polar and rectangular form

z^7 + z^6 + z^5 + z^4 + z^3 + z^2 + z = 0

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