If A z 1- B z 2 and C z 2 are the three points in argand plane where - z 1- z 2- z 1- z 2- and -1- i z 1- iz 3- z 1- z 3- z 1-then-A- A- B and C lie on a fixed circle with centre z2-z3-2B- A- B- C form right angle triangleC- A- B- C form an equilateral triangleD- A- B- C form an obtuse angle triangle

(z_1/z_2)=±π & |z_1+i(z_3 z_1)|=|z_1|+|z_3 z_1| ⇒(z_1/z_3 z_1)=π/2 So, centre of circle =(z_2+z_3/2) and ABC is a right angled triangle ...

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Byju's Answer

Standard XII

Mathematics

Properties of Modulus

If A z 1, B z...

Question

If $A (z_{1}), B (z_{2})$ and $C (z_{2})$ are the three points in argand plane where $| z_{1} + z_{2} | = | z_{1} | - | z_{2} |$ and $| (1 - i) z_{1} + i z_{3} | = | z_{1} | + | z_{3} - z_{1} |,$ then-

A, B

and

C

lie on a fixed circle with centre

(\frac{z_{2} + z_{3}}{2})

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A, B, C

form right angle triangle

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A, B, C

form an equilateral triangle

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A, B, C

form an obtuse angle triangle

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Solution

The correct options are
A $A, B$ and $C$ lie on a fixed circle with centre $(\frac{z_{2} + z_{3}}{2})$
B $A, B, C$ form right angle triangle
$arg (\frac{z_{1}}{z_{2}}) = \pm π & | z_{1} + i (z_{3} - z_{1}) | = | z_{1} | + | z_{3} - z_{1} | \Rightarrow arg (\frac{z_{1}}{z_{3} - z_{1}}) = \frac{π}{2}$
So, centre of circle $= (\frac{z_{2} + z_{3}}{2})$
and $△ A B C$ is a right angled triangle

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Similar questions

If A(z₁), B(z₂) and C(z₃) are three points in the argand plane where

| z₁+ z₂ | = | |z₁| - |z₂| | and | (1 - i)z₁+ iz₃ | = | z₁| + | z₃- z₁| ,

where i = √( -1) then

(a) A, B and C lie on a fixed circle with centre (z₂+ z₃)/2

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If A(z₁), B(z₂) and C(z₃) are three points on the argand plane where |z₁ + z₂| = ||z₁| - |z₂|| and |(1 - i)z₁ + iz₃| = |z₁| + |z₃ - z₁|,

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Q. If

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C

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

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z_{1}, z_{2}, z_{3}

be three non-zero complex numbers such that

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∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ = 0

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If A(z1),B(z2) and C(z2) are the three points in argand plane where |z1+z2|=|z1|−|z2| and |(1−i)z1+iz3|=|z1|+|z3−z1|, then-

The correct options are A A,B and C lie on a fixed circle with centre (z2+z32) B A,B,C form right angle trianglearg(z1z2)=±π & |z1+i(z3−z1)|=|z1|+|z3−z1|⇒arg(z1z3−z1)=π2 So, centre of circle =(z2+z32) and △ABC is a right angled triangle

If $A (z_{1}), B (z_{2})$ and $C (z_{2})$ are the three points in argand plane where $| z_{1} + z_{2} | = | z_{1} | - | z_{2} |$ and $| (1 - i) z_{1} + i z_{3} | = | z_{1} | + | z_{3} - z_{1} |,$ then-