Question

Find the realation between x and y of a complex variable z=x+iy satisfying the condition |z-3|/|z+3|=2

Answer 1

Answer:

Step-by-step explanation:

We have,

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

$\sf{\bullet\,\red{squarring\,\,both\,\,sides}}$

$\tt{\dfrac{|z-3|^2}{|z+3|^2}=(2)^2}$

$\tt{\implies\dfrac{|x+iy-3|^2}{|x+iy+3|^2}=(2)^2}$

$\tt{\implies\dfrac{|(x-3)+iy|^2}{|(x+3)+iy|^2}=(2)^2}$

$\tt{\implies\dfrac{(x-3)^2+y^2}{(x+3)^2+y^2}=4}$

$\tt{\implies(x-3)^2+y^2=4\{(x+3)^2+y^2\}}$

$\tt{\implies(x-3)^2+y^2=4(x+3)^2+4y^2}$

$\tt{\implies\,x^2-6x+9+y^2=4(x^2+6x+9)+4y^2}$

$\tt{\implies\,x^2+y^2-6x+9=4x^2+4y^2+24x+36}$

$\tt{\implies\,4x^2+4y^2+24x+36-x^2-y^2+6x-9=0}$

$\tt{\implies\,3x^2+3y^2+30x+27=0}$

$\tt{\implies\,x^2+y^2+10x+9=0}$