Question

State True or 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).

Options

• True

• False

Answer 1

This statement is True.

Explanation:

Let z = x + yi

Given that: |z – 1| = |z – i|

Then |z + yi – 1| = |x + yi – i|

⇒ `|(x - 1) + yi| = |x - (1 - y)i|`

⇒ `sqrt((x - 1)^2 + y^2) = sqrt(x^2 + (1 - y^2))`

⇒ (x – 1)² + y² = x² + (1 – y)²

⇒ x² – 2x + 1 + y² = x² + 1 + y² – 2y

⇒ –2x + 2y = 0

⇒ x – y = 0

Which is a straight line.

Slope = 1

Now equation of a line through the point (1, 0) and (0, 1).

y – 0 = `(1 - 0)/(0 - 1) (x - 1)`

⇒ y = –x + 1 whose slope = –1.

Now the multiplication of the slopes of two lines = –1 × 1 = –1

So they are perpendicular.