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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
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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 + 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.
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