Question

The complex numbers z = x + iy which satisfy the equation $\left| {\dfrac{{{\text{z - 5i}}}}{{{\text{z + 5i}}}}} \right|{\text{ = 1}}$ lie on:
A. the x - axis
B. straight line y = 5
C. a circle through the origin
D. none of these

Answer 1

Hint: To solve this problem we will use the property of modulus of complex numbers and use given conditions to create equations to find the solution.

Complete step-by-step answer:
Now, we are given a complex number z, where z = x + iy. Now, we will apply the property of complex numbers. We will use the property: $\left| {{\text{x + iy}}} \right|{\text{ }} = {\text{ }}\sqrt {{{\text{x}}^2}{\text{ + (iy}}{{\text{)}}^2}} $.
We will put the value of z in the given condition $\left| {\dfrac{{{\text{z - 5i}}}}{{{\text{z + 5i}}}}} \right|{\text{ = 1}}$ and also apply the above property.
Now, putting z = x + iy in the given condition, we get
$\left| {\dfrac{{{\text{x + iy - 5i}}}}{{{\text{x + iy + 5i}}}}} \right|{\text{ = 1}}$
Now, separating the real part and imaginary part, we get
$\left| {\dfrac{{{\text{x + i(y - 5)}}}}{{{\text{x + i(y + 5)}}}}} \right|{\text{ = 1}}$
Cross - multiplying both sides, we get
$\left| {{\text{x + i(y - 5)}}} \right|{\text{ = }}\left| {{\text{x + i(y + 5)}}} \right|$
Using the property $\left| {{\text{x + iy}}} \right|{\text{ }} = {\text{ }}\sqrt {{{\text{x}}^2}{\text{ + (iy}}{{\text{)}}^2}} $ in the above equation,
$\sqrt {{{\text{x}}^2}{\text{ + (i(y - 5)}}{{\text{)}}^2}} = {\text{ }}\sqrt {{{\text{x}}^2}{\text{ + (i(y + 5)}}{{\text{)}}^2}} $
$\sqrt {{{\text{x}}^2}{\text{ - (y - 5}}{{\text{)}}^2}} = {\text{ }}\sqrt {{{\text{x}}^2}{\text{ - (y + 5}}{{\text{)}}^2}} $ as ${{\text{i}}^2}{\text{ = - 1}}$
Squaring both sides, we get
${{\text{x}}^2}{\text{ - (y - 5}}{{\text{)}}^2}{\text{ = }}{{\text{x}}^2}{\text{ - (y + 5}}{{\text{)}}^2}$
Eliminating same terms from both the sides of the above equation, we get
${{\text{(y - 5)}}^2} = {\text{ (y + 5}}{{\text{)}}^2}$
${{\text{(y - 5)}}^2}{\text{ - (y + 5}}{{\text{)}}^2}{\text{ = 0}}$
Using property ${{\text{a}}^2}{\text{ - }}{{\text{b}}^2}{\text{ = (a - b)(a + b)}}$ in the above equation,
$({\text{y - 5 - y + 5)( y - 5 + y + 5) = 0}}$ which gives
-10y = 0 or y = 0
Now, y = 0 represents the x – axis. So, all the complex numbers satisfying $\left| {\dfrac{{{\text{z - 5i}}}}{{{\text{z + 5i}}}}} \right|{\text{ = 1}}$ lies on the x – axis.
So, option (A) is the correct answer.

Note: Such types of problems in which there is a condition which includes the complex number and asks the locus or the path of complex numbers the easiest method is to put the value of complex number x + iy wherever z is written, then solve the equation accordingly to get the desired answer.