Question

The locus represented by $ \left| {z - 1} \right| = \left| {z + i} \right| $ is:
A. a circle of radius 1 unit
B. An ellipse with foci at (1, 0) and (0, 1)
C. A straight line through the origin
D. A circle on the line joining (1, 0) and (0, 1) as diameter

Answer 1

Hint: z is a complex number. A complex number has both a real part and an imaginary part. A complex number is represented in an ordered pair in the complex plane. The general form every complex number, z, is $ z = x + iy $ , where i is the square root of -1 and the absolute value of a complex number, $ \left| {x + iy} \right| $ , is equal to $ \sqrt {{x^2} + {y^2}} $ . So in the given equation replace z with $ x + iy $ , and find its absolute values to find the locus.

Complete step by step solution:
We are given to find the locus represented by the line $ \left| {z - 1} \right| = \left| {z + i} \right| $
Let z be $ x + iy $
On replacing z in the equation $ \left| {z - 1} \right| = \left| {z + i} \right| $ with $ x + iy $ , we get
 $ \left| {x + iy - 1} \right| = \left| {x + iy + i} \right| $
Putting the terms without i together and with i together, we get
 $ \left| {x - 1 + iy} \right| = \left| {x + i\left( {y + 1} \right)} \right| $
Write the absolute values of the above complex numbers
 $ \sqrt {{{\left( {x - 1} \right)}^2} + {y^2}} = \sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} $
On squaring on both sides, we get
 $ \Rightarrow {\left( {x - 1} \right)^2} + {y^2} = {x^2} + {\left( {y + 1} \right)^2} $
We know the value of $ {\left( {a + b} \right)^2} $ is $ {a^2} + 2ab + {b^2} $ and the value of $ {\left( {a - b} \right)^2} $ is $ {a^2} - 2ab + {b^2} $
 $ \Rightarrow {x^2} - 2x\left( 1 \right) + {1^2} + {y^2} = {x^2} + {y^2} + 2y\left( 1 \right) + {1^2} $
 $ \Rightarrow {x^2} - 2x + 1 + {y^2} = {x^2} + {y^2} + 2y + 1 $
 $ \Rightarrow - 2x = 2y $
 $ \Rightarrow 2x + 2y = 0 $
 $ \Rightarrow x + y = 0 $
Therefore, the above resulted line is a straight line. When we substitute x as 0 then y will also be 0. This means that the line $ x + y = 0 $ is a straight line and passes through origin.
So, the correct answer is “Option C”.

Note: The absolute value of z is the distance of z from 0 in the complex plane. As the complex number can be represented in an ordered pair (x, y), distance from point (x, y) to 0 (0, 0) is the absolute value of z. Real numbers are all the numbers on the number line but complex numbers have a plane with an additional real axis to calculate the square roots of negative numbers.