Question

If we have an expression $\left| {\dfrac{{z - 25}}{{z - 1}}} \right| = 5$, then find the value of $\left| z \right|$
1) $3$
2) $4$
3) $5$
4) $6$

Answer 1

Hint: We know that if two complex numbers ${z_1}$ and ${z_2}$ are in the form $\left| {\dfrac{{{z_1}}}{{{z_2}}}} \right|$, then these can be written as $\dfrac{{\left| {{z_1}} \right|}}{{\left| {{z_2}} \right|}}$. Using this property in the given condition we can simplify the complex equation. Then, we know, if $z = x + iy$, then $\left| z \right| = \sqrt {{x^2} + {y^2}} $. Further using this property, the answer can be easily deduced.

Complete step-by-step solution:
Given, $\left| {\dfrac{{z - 25}}{{z - 1}}} \right| = 5$.
Now, we have to find value of the expression $\left| z \right|$.
Let us assume, $z = x + iy$.
Also, given, $\left| {\dfrac{{z - 25}}{{z - 1}}} \right| = 5$
We know, we can split the modulus function over division.
Therefore, $\dfrac{{\left| {z - 25} \right|}}{{\left| {z - 1} \right|}} = 5$
Now, multiplying both sides with $\left| {z - 1} \right|$, we get,
$ \Rightarrow \left| {z - 25} \right| = 5\left| {z - 1} \right|$
Substituting the value of $z$ that we assumed, we get,
$ \Rightarrow \left| {\left( {x + iy} \right) - 25} \right| = 5\left| {\left( {x + iy} \right) - 1} \right|$
Now, opening the brackets, we get,
$ \Rightarrow \left| {x + iy - 25} \right| = 5\left| {x + iy - 1} \right|$
Taking the real parts together, we get,
$ \Rightarrow \left| {\left( {x - 25} \right) + iy} \right| = 5\left| {\left( {x - 1} \right) + iy} \right|$
Now, we know, $\left| z \right| = \sqrt {{x^2} + {y^2}} $.
Therefore, using this property, we get,
$ \Rightarrow \sqrt {{{\left( {x - 25} \right)}^2} + {{\left( y \right)}^2}} = 5\sqrt {{{\left( {x - 1} \right)}^2} + {{\left( y \right)}^2}} $
Using, the formula ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$, we get,
$ \Rightarrow \sqrt {\left( {{x^2} - 50x + 625} \right) + {y^2}} = 5\sqrt {\left( {{x^2} - 2x + 1} \right) + {y^2}} $
Now, squaring both sides, we get,
$ \Rightarrow \left( {{x^2} - 50x + 625} \right) + {y^2} = 25\left\{ {\left( {{x^2} - 2x + 1} \right) + {y^2}} \right\}$
Opening the brackets and simplifying, we get,
$ \Rightarrow {x^2} - 50x + 625 + {y^2} = 25{x^2} - 50x + 25 + 25{y^2}$
Now, cancelling the similar terms, gives us,
$ \Rightarrow {x^2} + 625 + {y^2} = 25{x^2} + 25 + 25{y^2}$
Subtracting both sides by ${x^2}$ and ${y^2}$, gives us,
$ \Rightarrow 625 = 25{x^2} - {x^2} + 25 + 25{y^2} - {y^2}$
Now, subtracting both sides by $25$, we get,
$ \Rightarrow 625 - 25 = 24{x^2} + 24{y^2}$
$ \Rightarrow 600 = 24{x^2} + 24{y^2}$
Dividing, both sides by $24$, we get,
$ \Rightarrow 25 = {x^2} + {y^2}$
Now, we know, $\left| z \right| = \sqrt {{x^2} + {y^2}} $
$ \Rightarrow {\left| z \right|^2} = {x^2} + {y^2}$
Substituting this in the above equation, we get,
$ \Rightarrow 25 = {\left| z \right|^2}$
Changing the sides, gives us,
$ \Rightarrow {\left| z \right|^2} = 25$
Now, using square root on both sides, gives us,
$ \Rightarrow \left| z \right| = 5$
We will take only the positive value, as modulus function is always positive.
Therefore, the value of $\left| z \right|$ is $5$, the correct option is 3.

Note: The complex numbers consist of two parts, imaginary and real parts and are written in the form of $a + ib$, where is the real part and $ib$ is the imaginary part and $i $(called iota) has the value $\sqrt{-1}$ . We should take care of the calculations so as to be sure of the final answer. One must know the properties of the modulus of complex numbers to be able to do such questions. $a$