Let O be the origin and A be the point z 1=1+2 i. If B is the point z 2, operatornameRe( z 2)

Question

Let O be the origin and A be the point z 1=1+2 i. If B is the point z 2, operatornameRe( 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=π- tan -1 3 (B)  arg ( z 1-2 z 2)=- tan -1 (4/3) (C) |z2|=√10 (D) |2 z1-z2|=5

Tardigrade Admin · Accepted Answer

AB = AO ⋅ z -i π / 2=-2+i So OB =(-2+i)+(1+2 i) z 2=-1+3 i ∴|2 z 1- z 2|=√10

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

$ar g z_{2} = π - tan^{- 1} 3$

$ar g (z_{1} - 2 z_{2}) = - tan^{- 1} \frac{4}{3}$

$∣ z_{2} ∣ = 10$

$∣ 2 z_{1} - z_{2} ∣ = 5$

$A B = A O \cdot z^{- iπ /2} = - 2 + i$
So $OB = (- 2 + i) + (1 + 2 i)$
$z_{2} = - 1 + 3 i$
$∴ ∣ 2 z_{1} - z_{2} ∣ = 10$

Q. Let O be the origin and A be the point z1​=1+2i. If B is the point z2​,Re(z2​)<0, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true ?

argz2​=π−tan−13

arg(z1​−2z2​)=−tan−134​

∣z2​∣=10​

∣2z1​−z2​∣=5