Question

Among the complex numbers $z$ which satisfy the condition $|z-25i|\le 15,$ find the number having the least positive and greatest positive argument.

Answer 1

The complex numbers $z$ satisfying the condition
$|z-25i|\le 15$
are represented by the points inside and
on this circle of radius $15$ and centre at the point $C(0,25)$. From the figure, it is clear that the complex number $z$ satisfying $\left(1\right)$ and having least positive argument correspond to the point $P(x,y)$ which is the point of contact
ofa ray issuing from the origin and lying in the first quadrant
to the above circle. The positive argument of all other points
which and on the circle is greater than the argument of $P$
From figure, we have
$OC=25,CP=radius=15$
and $\angle CPO=90$
Hence $OP=\sqrt{{OC}^2-{CP}^2}=\sqrt{{25}^2-{15}^2}=20$.
If $\angle PCO=\theta$, then $\angle PON=\theta$, Also
$cos\theta =\frac{PC}{OC}=\frac{15}{25}=\frac{3}{5},$
$\therefore x=ON=OP\cos \theta =20\times \left(3/5\right)=12,$
and $y=PN=OP\sin \theta =20\times \left(4/5\right)=16,$.
Hence $P$ represents the complex number $z=x+iy=12+16i$ which
is therefore the required value of $z$ satisfying the given
conditions. The corresponding point $Q$ on the other tangent
from $O$ will have greatest $+ive$ argument. It will be $-12+16i$ whose argument is $\pi -\theta$.