If Rez−$$12z+i=1$$, where z $$=$$ x $$+$$ iy, then the point (x$$, $$y) lies on a

Correct option is 4circle whose centre is at −12,−34

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If $Re (\frac{z - 1}{2 z + i}) = 1$ , where z = x + iy, then the point (x, y) lies on a

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straight line whose slope is −23

circle whose diameter is 54

straight line whose slope is 32

circle whose centre is at −12,−34

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detailed solution

Correct option is D

Let z=x+iy. Thenz−12z+i=x−1+iy2x+iy+i=x−1+iy2x+2y−1i×2x−2y+1i2x−2y+1Now Rez+12z+i=2xx−1+y2y+12x2+2y+12=1⇒2x2+2y2−2x+y=4x2+4y2+4y+1⇒2x2+2y2+2x+3y+1=0⇒x2+y2+x+32y+12=0circle with centre −12,−34r=14+916−12=4+9−816=54

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If Rez−12z+i=1, where z = x + iy, then the point (x, y) lies on a

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If $Re (\frac{z - 1}{2 z + i}) = 1$ , where z = x + iy, then the point (x, y) lies on a