 Questions & Answers    Question Answers

# The complex number $z = x + iy$ which satisfy the equation $\left| {\dfrac{{z - 5i}}{{z + 5i}}} \right| = 1$, lie on${\text{A}}{\text{.}}$ The x-axis${\text{B}}{\text{.}}$ The straight line $y = 5$${\text{C}}{\text{.}}$ A circle passing through the origin${\text{D}}{\text{.}}$ None of these.  Answer Verified
Hint – In this question use the property of modulus of a complex number which is $\left| {A + iB} \right| = \sqrt {{A^2} + {B^2}}$ to reach the answer.

Given equation is
$\left| {\dfrac{{z - 5i}}{{z + 5i}}} \right| = 1$, where $z = x + iy$
Now as we know $\left| {\dfrac{A}{B}} \right| = \dfrac{{\left| A \right|}}{{\left| B \right|}}$
$\Rightarrow \left| {\dfrac{{z - 5i}}{{z + 5i}}} \right| = \dfrac{{\left| {z - 5i} \right|}}{{\left| {z + 5i} \right|}} = 1$
$\Rightarrow \left| {z - 5i} \right| = \left| {z + 5i} \right|$
Now substitute $z = x + iy$
$\Rightarrow \left| {x + iy - 5i} \right| = \left| {x + iy + 5i} \right| \\ \Rightarrow \left| {x + i\left( {y - 5} \right)} \right| = \left| {x + i\left( {y + 5} \right)} \right| \\$
Now as we know that $\left| {A + iB} \right| = \sqrt {{A^2} + {B^2}}$, so use this property we have
$\sqrt {{x^2} + {{\left( {y - 5} \right)}^2}} = \sqrt {{x^2} + {{\left( {y + 5} \right)}^2}}$
Now squaring on both sides we have
${x^2} + {\left( {y - 5} \right)^2} = {x^2} + {\left( {y + 5} \right)^2} \\ \Rightarrow {\left( {y - 5} \right)^2} = {\left( {y + 5} \right)^2} \\$
Now opening the square we have
${y^2} + 25 - 10y = {y^2} + 25 + 10y \\ \Rightarrow 20y = 0 \\ \Rightarrow y = 0 \\$
And we all know y = 0 is nothing but a x-axis
Hence option (a) is correct.

Note – In such types of questions the key concept we have to remember is that always recall all the properties of modulus which is stated above, then according to these properties simplify the given equation we will get the required answer.
Bookmark added to your notes.
View Notes
Conjugate of a Complex Number  Solution of Quadratic Equation in Complex Number System  Complex Numbers  Argument of Complex Numbers  Polar Form of Complex Numbers  Algebraic Operations on Complex Numbers  Complex Numbers and Quadratic Equations  Complex Carbohydrates  Fractions on the Number Line  CBSE Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations Formulas  