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:
\(tan^{-1}(\frac {2}{\sqrt5})-\pi\)
\(tan^{-1}(\frac {24}{7})-\pi\)
\(tan^{-1}(3)-\pi\)
\(tan^{-1}(\frac 34)-\pi\)
The Correct Option is B
Solution and Explanation
\(z_1 = 3 + 4i\)
\(z_2 = 4 + 3i\)
\(z_3 = 5i\)
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): \(tan^{-1}(\frac {24}{7})-\pi\)
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.