Let z=x+iy ∴|z|=√x2+y2 Given, |z|=3 ∴√x2+y2=3 ⇒x2+y2=9=32 This represent a circle with center at (0,0) and radius =3 Now, given arg(z−1)−arg(z+1)=‌
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4
⇒arg(x+iy−1)−arg(x+iy+1)=‌
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4
⇒arg(x−1+iy)−arg(x+1+iy)=‌
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⇒tan−1(‌
y
x−1
)−tan−1(‌
y
x+1
)=‌
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⇒tan−1(‌
‌
y
x−1
−
y
x+1
1+‌
y
x−1
×‌
y
x+1
)=‌
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⇒tan−1(‌
‌
xy+y−xy+y
x2−1
x2−1+y2
x2−1
)=‌
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⇒tan−1(‌
xy+y−xy+y
x2−1+y2
)=‌
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⇒‌
2y
x2−1+y2
=tan(‌
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) ⇒2y=x2+y2−1 ⇒x2+y2−2y−1=0 ⇒x2+(y−1)2=(√2)2 This represent a circle with center at (0,1) and radius √2.
From diagram you can see both the circles do not cut anywhere.