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

Question

State True of False:

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

Answer 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.