Question

Let a circle $C$ in complex plane pass tltrough the points $z _{1}=3+4 i , z _{2}=4+3 i$ and $z _{3}=5 i$. If $z\left(\neq z_{1}\right)$ is a point on $C$ such that the line through $Z$ and $z _{1}$ is perpendicular to the line through $z _{2}$ and $z _{3}$, then $\arg ( z )$ is equal to :

• A. $\tan ^{-1}\left(\frac{2}{\sqrt{5}}\right)-\pi$
• B. $\tan ^{-1}\left(\frac{24}{7}\right)-\pi$
• C. $\tan ^{-1}(3)-\pi$
• D. $\tan ^{-1}\left(\frac{3}{4}\right)-\pi$

Answer 1

Answer: (B)

Slope of $BC =\frac{3-5}{4-0}=-\frac{1}{2}$
Slope of $AP =2$
equation of $AP : y-4=2(x-3)$
$\Rightarrow y =2( x -1)$
$P$ lies on circle $x^{2}+y^{2}=25$
$\Rightarrow x ^{2}+(2( x -1))^{2}=25 $
$\Rightarrow x =-\frac{7}{5} $ and $y =-\frac{24}{5} $
$\Rightarrow \arg ( z )=\tan ^{-1}\left(\frac{24}{7}\right)-\pi$