Let O be the origin and A be the point $\mathrm{z}_{1}=1+2 i$. 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?

(D) $\left|2 \mathrm{z}_{1}-\mathrm{z}_{2}\right|=5$

(A) $\arg \mathrm{z}_{2}=\pi-\tan ^{-1} 3$

(B) $\arg \left(z_{1}-2 z_{2}\right)=-\tan ^{-1} \frac{4}{3}$

(C) $\left|\mathrm{z}_{2}\right|=\sqrt{10}$

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Complex Numbers

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?

Answer: (D)

Since

O A B

is a right-angled isosceles triangle with right angle at

A

and

O B

as hypotenuse, we have

∣ O A ∣ = ∣ A B ∣

and

∠ O A B = 9 0^{\circ}

.

The vector

A B

is obtained by rotating

A O

- 9 0^{\circ}

A B = - z_{1} \cdot e^{- iπ /2} = - z_{1} \cdot (- i) = i z_{1} = i (1 + 2 i) = - 2 + i

.

Thus

z_{2} = z_{1} + A B = (1 + 2 i) + (- 2 + i) = - 1 + 3 i

.

Now check option (D):

∣2 z_{1} - z_{2} ∣ = ∣2 (1 + 2 i) - (- 1 + 3 i) ∣ = ∣2 + 4 i + 1 - 3 i ∣ = ∣3 + i ∣ = 10

, not 5.

All other options can be verified to be true.

JEE Main · 2022 · 26 Jul 2022 · Shift 1 · Single correct option