Question

Let O be the origin and A be the point \(z_1 = 1 + 2i\). If B is the point \(z_2, Re(z_2) < 0\), such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?

• A. \(arg\ z_2=\pi–tan^{−1}3\)
• B. \((arg)(z_1−2z_2)=−tan^{−1⁡}\frac 43\)
• C. \(|z_2|=\sqrt {10}\)
• D. \(|2z_1−z_2|=5\)

Answer 1

The Correct Option is D

\(\frac {z2−0}{(1+2i)−0}=\frac {|OB|}{|OA|}e^{\frac {i\pi}{4}}\)

\(⇒ \frac {z2}{1+2i}=\sqrt 2 e^{i\pi}{4}\)

\(z_2=(1+2i)(1+i)\)
\(z_2=−1+3i\)
\(arg\ z_2=π–tan^{−1}3\)
\(|z_2|=\sqrt {10}\)
\(z_1–2z_2=(1+2i)+2–6i\)
\(z_1–2z_2=3–4i\)
\(arg\ (z_1−2z_2)=−tan^{−1⁡}\frac 43\)
\(|2z_1−z_2|=|2+4i+1−3i|\)
\(|2z_1−z_2|=|3+i|\)
\(=\sqrt {10}\)

So, the correct option is (D): \(|2z_1−z_2|=5\)