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

The Correct Option is B

\(z_1 = 3 + 4i\)
\(z_2 = 4 + 3i\)
\(z_3 = 5i\)
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): \(tan^{-1}(\frac {24}{7})-\pi\)