Question

If $\dfrac{{\left| {z - 25} \right|}}{{\left| {z - 1} \right|}} = 5$, then find the value of $\left| z \right|$
A.3
B.4
C.5
D.6

Answer 1

Hint: First suppose that $z = x + iy$, then multiply $\left| {z - 1} \right|$ to both sides of the given equation and apply the modulus formula to remove the modulus. Then solve the equation ${(x - 25)^2} + {y^2} = 25\left[ {{{(x - 1)}^2} + {y^2}} \right]$ to obtain the required value.

Formula Used:
The modulus formula of a complex number $z = x + iy$ is,
${\left| z \right|^2} = {x^2} + {y^2}$

Complete step by step solution:
The given equation is,
$\dfrac{{\left| {z - 25} \right|}}{{\left| {z - 1} \right|}} = 5$
Multiply $\left| {z - 1} \right|$ to both sides of the given equation,
$\left| {z - 25} \right| = 5\left| {z - 1} \right|$
Square both sides of the equation $\left| {z - 25} \right| = 5\left| {z - 1} \right|$,
${\left| {z - 25} \right|^2} = 25{\left| {z - 1} \right|^2}$
Put $z = x + iy$ in the equation ${\left| {z - 25} \right|^2} = 25{\left| {z - 1} \right|^2}$
${\left| {x + iy - 25} \right|^2} = 25{\left| {x + iy - 1} \right|^2}$
Use the definition of modulus and proceed further
${\left( {x - 25} \right)^2} + {y^2} = 25\left[ {{{\left( {x - 1} \right)}^2} + {y^2}} \right]$
${x^2} - 50x + 625 + {y^2} = 25\left[ {{x^2} - 2x + 1 + {y^2}} \right]$
${x^2} - 50x + 625 + {y^2} = 25{x^2} - 50x + 25 + 25{y^2}$
$25{x^2} - {x^2} + 25{y^2} - {y^2} = 625 - 25$
$24({x^2} + {y^2}) = 600$
${x^2} + {y^2} = 25$
${\left| z \right|^2} = {5^2}$
$\left| z \right| = 5$

Option ‘C’ is correct

Additional Information:
Complex numbers are those that are expressed as a+ib, where a and b are the real numbers and i is an imaginary number called an "iota." The complex number's modulus is the angle between the complex number's origin and the point it represents on the argand plane. Complex numbers with Real and Imaginary elements are greater than their negative modulus and less than their positive modulus.

Note: To solve this type of problem, you have to remember that z denotes a complex number. The most important part here is to multiply $\left| {z - 1} \right|$ on both sides of the equation and take square both sides of the equation. Then substitute $z = x + iy$ in the equation and calculate the value of $\left| z \right|$.