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MCQ

True or False

**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

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#### Solution

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} + y^{2} = x^{2} + (1 – y)^{2}

⇒ x^{2} – 2x + 1 + y^{2} = x^{2} + 1 + y^{2} – 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.

Concept: Algebraic Operations of Complex Numbers

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