Statement 1- Let z1 and z2 are two complex numbers such that -z1-z2-z1-z2- then the orthocenter of - AOB is z1-z22- where O is origin-Statement 2- In case of a right-angled triangle- orthocenter is that point at which the triangle is right-angled-

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Modulus of a Complex Number

Statement 1: ...

Question

Statement 1: Let $z_{1}$ and $z_{2}$ are two complex numbers such that $| z_{1} - z_{2} | = | z_{1} + z_{2} |$ , then the orthocenter of $Δ A O B$ is $\frac{(z_{1} + z_{2})}{2}$ . (where O is origin).
Statement 2: In case of a right-angled triangle, orthocenter is that point at which the triangle is right-angled.

Both the statements are true, and Statement 2 is the correct explanation for Statement 1.

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Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

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Statement 1 is true and Statement 2 is false.

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Statement 1 is false and Statement 2 is true.

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z_{1}

z_{2}

| z_{1} - z_{2} | = | z_{1} + z_{2} |

△ A O B

\frac{z_{1} + z_{2}}{2}

O

z_{1}

z_{2}

| z_{1} | = | z_{2} | + | z_{1} - z_{2} |

I m (\frac{z_{1}}{z_{2}}) = 0

a r g (z) = 0 \Rightarrow z

| z_{1} | = | z_{2} | = | z_{3} |, z_{1} + z_{2} + z_{3} = 0

(z_{1}), B (z_{2}), C (z_{3})

Δ A B C

a r g (\frac{(z_{2} + z_{3} - 2 z_{1})}{(z_{3} - z_{2})})

\frac{π}{2}

1 :

x y + y z + z x = 1

x, y, z \in R^{+}

\frac{x}{1 + x^{2}} + \frac{y}{1 + y^{2}} + \frac{z}{1 + z^{2}} = \frac{2}{\sqrt{(1 + x^{2}) (1 + y^{2}) (1 + z^{2})}}

2 :

A B C

sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C

z_{1}, z_{2}

\frac{z_{1} - {2 z}_{2}}{2 - z_{2} {¯ z}_{2}}

z_{2}

| z_{1} | = 2

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