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Question

If Re(z12z+i)=1, where z=x+iy, then the point (x,y) lies on a

A
circle whose centre is at (12,32).
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B
straight line whose slope is 32.
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C
circle whose diameter is 52.
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D
straight line whose slope is 23.
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Solution

The correct option is C circle whose diameter is 52.
z=x+iyx+iy12x+2iy+i=(x1)+iy2x+i(2y+1)(2xi(2y+1)2xi(2y+1))2x(x1)+y(2y+1)4x2+(2y+1)2=12x2+2y22x+y=4x2+4y2+4y+1x2+y2+x+32y+12=0

Circle's centre will be (12,34)

Radius =14+91612=54

Diameter =52

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