Question

The complex number $z$ having least positive argument which satisfies the condition $|z-25i|\le 15$

• A. $25i$
• B. $12+5i$
• C. $16+12i$
• D. $12+16i$

Answer 1

Answer: (A), (D)

The correct options are
A $25i$
D $12+16i$

Given that z has least positive argument
$\therefore arg\left(z\right)\ge 0$
$\therefore {\tan }^{-1}\left(\left|y/x\right|\right)\ge 0$
also $|z-25i|\le 15$
i.e $x^2+(y-25{)}^2\le 225$
This is the equation of all points lying on or inside the circle with center(0,25i) and radius $=$15
$\therefore$ as shown in diagram, the least +ve argument will be of the point on the circle whose tangent passes through origin
$arg\left(z\right)=\theta =90-{\sin }^{-1}\left(\frac{15}{25}\right)$
$=90-{53}^{\circ }$
$={37}^{\circ }$
$\left|z\right|=\sqrt{x^2+y^2}$
$\Rightarrow \left|z\right|=\sqrt{{25}^2-{15}^2}$
$=\sqrt{400}$
$=20$
$\therefore ans=12+16i$