Question

Let a circle C in complex plane pass through the points $z_1 = 3 + 4i$, $z_2 = 4 + 3i$ and $z_3 = 5i$. If $z(≠z_1)$ 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. $tan^{-1}(\frac {2}{\sqrt5})-\pi$
• B. $tan^{-1}(\frac {24}{7})-\pi$
• C. $tan^{-1}(3)-\pi$
• D. $tan^{-1}(\frac 34)-\pi$

Answer 1

Answer: (B)

$tan^{-1}(\frac {24}{7})-\pi$

Answer 2

$z_1 = 3 + 4i$
$z_2 = 4 + 3i$
$z_3 = 5i$
Clearly,
$C = x^2 + y^2 = 25$
Let z(x, y)
$(\frac {y−4}{y−3})(\frac {2}{−4})=−1$
$y = 2x – 2 = L$
So, z is intersection of C&L
$z=(−\frac 75,−\frac {24}{5})$
Therefore, Arg(z) $=tan^{-1}(\frac {24}{7})-\pi$

So, the correct option is (B): $tan^{-1}(\frac {24}{7})-\pi$