If z 1- z 2- z 3 be three complex numbers which are in H-P- And the points A z 1- B z 2- C z 3 are non-collinear- and O is origin- then-A- O- A- B- C are concyclicB- O is centroid of - A B CC- O- A- B- C are collinearD- None of these

The correct option is A O, A, B, C are concyclic z_2 = 2z_1z_3/z_1+z_3 z_2 0/z_1 0 = | z_2/z_1 | e^iα | z_3/z_1 z_3 | e^iβ = 1/2| z_2/z_1 | e^iβ ∴ arg z_2/z_1 ...

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Byju's Answer

Standard XI

Mathematics

Slope of a Line Connecting Two Points

If z 1, z 2, ...

Question

If $z_{1}, z_{2}, z_{3}$ be three complex numbers which are in H.P. And the points $A (z_{1}), B (z_{2}), C (z_{3})$ are non-collinear, and O is origin, then:

O is centroid of ∆ ABC

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O, A, B, C are concyclic

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O, A, B, C are collinear

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None of these

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Solution

The correct option is B
O, A, B, C are concyclic

$z_{2}$ = $\frac{2 z_{1} z_{3}}{z_{1} + z_{3}}$

$\frac{z_{2} - 0}{z_{1} - 0}$ = $∣ ∣ \frac{z_{2}}{z_{1}} ∣ ∣ e^{i α}$

$∣ ∣ \frac{z_{3}}{z_{1} - z_{3}} ∣ ∣ e^{i β}$ = $\frac{1}{2}$ $∣ ∣ \frac{z_{2}}{z_{1}} ∣ ∣ e^{i β}$

$∴ a r g ⎧ ⎩ \frac{z_{2}}{z_{1}} ⎫ ⎭$ = $a r g ⎧ ⎩ \frac{z_{2} - z_{3}}{z_{1} - z_{3}} ⎫ ⎭$

$\Rightarrow ∠ A O B$ = $∠ A C B$

Hence, O,A,B,C are concyclic.

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If z1,z2,z3 be three complex numbers which are in H.P. And the points A(z1),B(z2),C(z3) are non-collinear, and O is origin, then:

The correct option is B O, A, B, C are concyclic z2 = 2z1z3z1+z3 z2−0z1−0 = ∣∣z2z1∣∣eiα ∣∣z3z1−z3∣∣eiβ = 12∣∣z2z1∣∣eiβ ∴arg ⎧⎩z2z1⎫⎭ = arg⎧⎩z2−z3z1−z3⎫⎭ ⇒∠AOB = ∠ACB Hence, O,A,B,C are concyclic.

If $z_{1}, z_{2}, z_{3}$ be three complex numbers which are in H.P. And the points $A (z_{1}), B (z_{2}), C (z_{3})$ are non-collinear, and O is origin, then: