Question

If $z$ satisfies $|z-25i|\le 15$, then which of the following is false statement.
(i) Least positive argument.
(ii) Maximum positive argument.
(iii)Least modulus
(iv) greatest modulus

• A. Least positive argumen$t=ta{n}^{-1}\frac{4}{3}$
• B. Maximum positive argumen$t=ta{n}^{-1}\frac{4}{3}$
• C. Least modulus$=10$
• D. greatest modulus$=40$

Answer 1

The correct option is B Maximum positive argumen$t=ta{n}^{-1}\frac{4}{3}$

The complex numbers z satisfying the condition
$|z-25i|\le 15$ ....... (i)
are represented by the points inside and on the circle of radius $15$ and centre at the point $C(0,25)$.
The complex number having least positive argument and maximum positive arguments in this region are the points of contact of tangents drawn from origin to the circle.
Here $\theta =$ least positive argument
and $\varphi =$ maximum positive argument
$\therefore$ In $\Delta OCP=OP=\sqrt{(OC{)}^2-(CP{)}^2}\sqrt{(25{)}^2-(15{)}^2}=20$
and $sin\theta =\frac{OP}{OC}=\frac{20}{25}=\frac{4}{5}$
$\therefore tan\theta =\frac{4}{3}\Rightarrow \theta =ta{n}^{-1}\left(\frac{4}{3}\right)$
Thus complex number at $P$ has modulus $20$ and argument $\Theta =ta{n}^{-1}\left(\frac{4}{3}\right)$
$\therefore Z_P=20(cos\theta +isin\theta )=20(\frac{3}{5}+i\frac{4}{5})$
$\therefore Z_P=12+16i$
Similarly $Z+Q=-12+16i$
From the figure, $E$ is the point with least modulus and $D$ is the point with maximum modulus.
Hence $Z_E=\overrightarrow{OE}=\overrightarrow{OC}-\overrightarrow{EC}=25i-15i=10i$
and $Z_D=\overrightarrow{OD}=\overrightarrow{Oc}+\overrightarrow{CD}=25i+15i=40i$
Also $|\text{Max amp z - Min amp z}|=|\pi -\theta -\theta |$
$=|\pi -2ta{n}^{-1}\frac{4}{3}|$
$=\pi -2ta{n}^{-1}\frac{4}{3}$
Ans: B