Question

Let a circle $C$ in complex plane pass tltrough the points $z_1=3+4i,z_2=4+3i$ and $z_3=5i$. If $z\left(\ne z_1\right)$ 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 $y=-\frac{24}{5}$
$\Rightarrow \arg (z)={\tan }^{-1}\left(\frac{24}{7}\right)-\pi$