Question

The number of solutions for the equations \[|z - 1| = |z - 2| = |z - i|\]is [Orissa JEE\[2005\]]
A) One solution
B) \[3\] Solution
C) \[2\] Solution
D) No solution

Answer 1

Hint: in this question we have to find which total number of solution of given equation. First, write the given complex number as a combination of real and imaginary numbers. Put z in form of real and imaginary number into the equation.

Formula Used: Equation of complex number is given by
\[z = x + iy\]
Where
z is a complex number
x represent real part of complex number
iy is a imaginary part of complex number
i is iota
Square of iota is equal to the negative of one

Complete step by step solution: First equation we have \[|z - 1| = |z - 2|\]
We know that complex number is written as a combination of real and imaginary number.
\[z = x + iy\]
Put this value in \[|z - 1| = |z - 2|\]
\[|(x + iy) - 1| = |(x + iy) - 2|\].
\[|(x - 1) + iy| = |(x - 2) + iy|\]
We know that
\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
\[\sqrt {{{(x - 1)}^2} + {y^2}} = \sqrt {{{(x - 2)}^2} + {y^2}} \]
\[{(x - 1)^2} + {y^2} = {(x - 2)^2} + {y^2}\]
On simplification we get
\[1 - 2x = 4 - 4x\]
\[2x = 3\]
\[x = \dfrac{3}{2}\]
Now we have equation
\[|z - 1| = |z - 2| = |z - i|\]
\[|z - 1| = |z - i|\]
We know that complex number is written as a combination of real and imaginary number.
\[z = x + iy\]
Put this value in \[|z - 1| = |z - i|\]
\[|(x + iy) - 1| = |(x + iy) - i|\]
\[|(x - 1) + iy| = |(x + i(y - 1)|\]
We know that modulus of complex number
\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
\[\sqrt {{{(x - 1)}^2} + {y^2}} = \sqrt {{x^2} + {{(y - 1)}^2}} \]
\[{(x - 1)^2} + {y^2} = {x^2} + {(y - 1)^2}\]
On simplification we get
\[1 - 2x = 1 - 2y\]
\[y = x\]
So only one solution is there for given complex equation

Option ‘A’ is correct

Note: Complex number is a number which is a combination of real and imaginary number. So in complex number questions, we have to represent number as a union of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to negative one.