Question

The complex numbers $z=x+iy$ which satisfy the equation $\frac{|z-5i|}{|z+5i|}=1$, lie on

• A. The x - axis
• B. The straight line y = 5
• C. A circle passing through the origin
• D. None of these

Answer 1

The correct option is A The x - axis

Given,

$\frac{|z-5i|}{|z+5i|}=1$

$\frac{|x+iy-5i|}{|x+iy+5i|}=1$

$\frac{|x+i(y-5)|}{|x+i(y+5)|}=1$

$\frac{\sqrt{x^2+(y-5{)}^2}}{\sqrt{x^2+(y+5{)}^2}}=1$

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

$x^2+(y-5{)}^2=x^2+(y+5{)}^2$

$-10y=10y$

$20y=0$

$y=0$