Let $O$ be the origin and $A$ be the point $z _1=1+2 i$. If $B$ is the point $z _2, \operatorname{Re}\left( z _2\right)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true ?
- $\arg z _2=\pi-\tan ^{-1} 3$
- $\arg \left( z _1-2 z _2\right)=-\tan ^{-1} \frac{4}{3}$
- $\left|z_2\right|=\sqrt{10}$
- $\left|2 z_1-z_2\right|=5$
The Correct Option is D
Solution and Explanation
The correct option is (D): $\left|2 z_1-z_2\right|=5$
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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.