Complex Numbers 1 Question 2

2. All the points in the set S=α+iαi:αR(i=1) lie on a

(2019 Main, 9 April I)

(a) circle whose radius is 2.

(b) straight line whose slope is -1 .

(c) circle whose radius is 1.

(d) straight line whose slope is 1 .

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Answer:

Correct Answer: 2. (c)

Solution:

  1. Let x+iy=α+iαi

x+iy=(α+i)2α2+1=(α21)+(2α)iα2+1=α21α2+1+2αα2+1i

On comparing real and imaginary parts, we get

x=α21α2+1 and y=2αα2+1

 Now, x2+y2=α21α2+12+2αα2+12=α4+12α2+4α2(α2+1)2=(α2+1)2(α2+1)2=1x2+y2=1

Which is an equation of circle with centre (0,0) and radius 1 unit.

So, S=α+iαi;αR lies on a circle with radius 1 .



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