Question

Which of the following equations can represent a triangle? [Orissa JEE\[2005\]]
A) \[|z - 1| = |z - 2|\]
B) \[|z - 1| = |z - 2| = |z - i|\]
C) \[|z - 1| - |z - 2| = 2a\]
D) \[|z - 1{|^2} + |z - 2{|^2} = 4\]

Answer 1

Hint: in this question we have to find which of the given equation represent a triangle. Check each given option. 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
\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
\[\sqrt {{{(x - 1)}^2} + {y^2}} = \sqrt {{{(x - 2)}^2} + {y^2}} \]
\[{(x - 1)^2} + {y^2} = {(x - 2)^2} + {y^2}\]
\[1 - 2x = 4 - 4x\]
\[2x = 3\] Equation of line………………………. (i)
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
\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
\[\sqrt {{{(x - 1)}^2} + {y^2}} = \sqrt {{x^2} + {{(y - 1)}^2}} \]
\[{(x - 1)^2} + {y^2} = {x^2} + {(y - 1)^2}\]
\[1 - 2x = 1 - 2y\]
\[y = x\]Equation of line…………………………. (ii)
Now we have equation \[|z - 2| = |z - i|\]

We know that complex number is written as a combination of real and imaginary number.
\[z = x + iy\]
Put this value in\[|z - 2| = |z - i|\]
\[|(x + iy) - 2| = |(x + iy) - i|\]
\[|(x - 2) + iy| = |x + i(y - 1)|\]
\[\sqrt {{{(x - 2)}^2} + {y^2}} = \sqrt {{x^2} + {{(y - 1)}^2}} \]
\[{(x - 2)^2} + {y^2} = {x^2} + {(y - 1)^2}\]
\[4 - 4x = 1 - 2y\]
\[4x - 2y = 3\] Equation of line………………………. (III)
From equation (i) , (ii) and (iii)
We can say that all three equations is equation of line and these three line form triangle

Option ‘B’ 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 combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to negative one.