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). - Mathematics

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).

• True

• False

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 + 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.

Concept: Algebraic Operations of Complex Numbers
Is there an error in this question or solution?

APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 26.(iv) | Page 93

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