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JEE Main 2017 · 9 Apr · Shift 1, Mathematics question 63

The equation Im⁡(iz−2z−i)+1=0\operatorname{Im}\left(\dfrac{iz-2}{z-i}\right)+1=0, z∈Cz\in\mathbb C, z≠iz\ne i represents a part of a circle having radius equal to :

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Answer and worked solution

Official answer: C

With z=x+iyz=x+iy,
Im⁡iz−2z−i=x2+y2+y−2x2+(y−1)2.\operatorname{Im}\frac{iz-2}{z-i}=\frac{x^2+y^2+y-2}{x^2+(y-1)^2}.
Setting it to −1 gives 2x2+2y2−y−1=02x^2+2y^2-y-1=0, i.e. x2+y2−y2−12=0x^2+y^2-\frac y2-\frac12=0, a circle with r2=116+12=916r^2=\frac1{16}+\frac12=\frac9{16}: r=34r=\frac34.

Topics:Geometry in the complex planeEquation of a circle
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