Let $z_1=2+3 i$ and $z_2=3+4 i$ The set $S=\left\{z \in C:\left|z-z_1\right|^2-\left|z-z_2\right|^2=\left|z_1-z_2\right|^2\right\}$ represents a
- straight line with the sum of its intercepts on the coordinate axes equals $-18$
- hyperbola with eccentricity 2
- hyperbola with the length of the transverse axis 7
- straight line with the sum of its intercepts on the coordinate axes equals 14
The Correct Option is D
Solution and Explanation
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Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
Then consider:
Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
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If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \) and
\[ \arg(z) - \arg(\omega) = \frac{\pi}{2}, \]
Then the value of \( \overline{z\omega} \) is:
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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.