Q26 of 66 Page 91

State True of False for the following:

The locus represented by |z – 1| = |z – i| is a line perpendicular to the join of (1, 0) and (0, 1).

True

Explanation:


Given |z – 1| = |z – i|


Putting z = x + iy,


⇒ |x – 1 + iy| = |x – i (1 – y) |


⇒ (x – 1)2 + y2 = x2 + (1 – y)2


⇒ x2 - 2x + 1 + y2 = x2 + 1 + y2 – 2y


⇒ -2x + 1 = 1 – 2y


⇒ -2x + 2y = 0


⇒ x – y = 0


Now, equation of a line through the points (1, 0) and (0, 1) is



⇒ x + y = 1


This line is perpendicular to the line x – y = 0.


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