The equation arg left(frac{z-1}{z+1}right)=fr | vedclass

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(B) Let $z = x + iy$. The equation $\arg \left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}$ represents the locus of a point $z$ such that the angle subtend

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centre at $(0,1)$ and radius $\sqrt{2}$

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(B) Let $z = x + iy$. The equation $\arg \left(\frac{z-1}{z+1}ight) = \frac{\pi}{4}$ represents the locus of a point $z$ such that the angle subtended by the segment joining $A(1, 0)$ and $B(-1, 0)$ at $z$ is $\frac{\pi}{4}$.This is an arc of a circle passing through $A(1, 0)$ and $B(-1, 0)$.Let the centre of the circle be $C(0, k)$. Since the angle at the circumference is $\frac{\pi}{4}$,the angle subtended by the chord $AB$ at the centre $C$ is $2 	imes \frac{\pi}{4} = \frac{\pi}{2}$.In $	riangle OAC$ (where $O$ is the origin $(0,0)$),$\angle COA = 90^\circ$ and $\angle OCA = \frac{\pi}{4}$.Since $OA = 1$,we have $	an \left(\frac{\pi}{4}ight) = \frac{OA}{OC} = \frac{1}{OC} = 1$,which implies $OC = 1$.Thus,the centre is $C(0, 1)$.The radius $R = AC = \sqrt{OA^2 + OC^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$.Therefore,the circle has centre $(0, 1)$ and radius $\sqrt{2}$.

The equation $\arg \left(\frac{z-1}{z+1}\right)=\frac{\pi}{4}$ represents a circle with:

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