Question
The complex numbers $$z = x+ iy$$ which satisfy the equation $$\left| {\frac{{z - 5i}}{{z + 5i}}} \right| = 1$$ lie on
A.
the $$x$$ - axis
B.
the straight line $$y = 5$$
C.
a circle passing through the origin
D.
none of these
Answer :
the $$x$$ - axis
Solution :
$$\eqalign{
& {\text{ATQ }}\left| {x + iy - 5i} \right| = \left| {x + iy + 5i} \right| \cr
& \Rightarrow \,\,\left| {x + \left( {y - 5} \right)i} \right| = \left| {x + \left( {y + 5} \right)i} \right| \cr
& \Rightarrow \,\,{x^2} + {\left( {y - 5} \right)^2} = {x^2} + {\left( {y + 5} \right)^2} \cr
& \Rightarrow \,\,{x^2} + {y^2} - 10y + 25 = {x^2} + {y^2} + 10y + 25 \cr
& \Rightarrow \,\,20y = 0 \cr
& \Rightarrow \,\,y = 0 \cr} $$
$$\therefore \,\,'A'$$ is the correct alternative.