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Distance Formula

If A z 1, B z...

Question

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|,

where i= √(- 1) then

(a) A, B and C lie on the fixed circle with centre (z₁ + z₂)/2

(b) A, B and C are collinear points

(c) ABC form an equilateral triangle

(d) ABC form an obtuse angled triangle

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Solution

Observe the graph

since z1 z2 and z3 are colinear its center is z1+z2/2

So if you are to draw a circle Then A,B,C will lie on that circl

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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

(b) A, B and C are collinear points

(c) ABC form an equilateral triangle

(d) ABC form an obtuse angle triangle

A (z_{1}), B (z_{2})

C (z_{2})

are the three points in argand plane where

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

| (1 - i) z_{1} + i z_{3} | = | z_{1} | + | z_{3} - z_{1} |,

z_{1}, z_{2}

z_{3}

are the affixes of three points

A, B

C

respectively and satisfy the condition

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

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

then prove that

△ A B C

in a right angled.

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

are the vertices of the triangle

A B C

(in anticlockwise order). If

∠ A B C = \frac{π}{4}

A B = \sqrt{2} (B C)

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

be three complex numbers and

a, b, c

be real numbers not all zero, such that

a + b + c = 0

a z_{1} + b z_{2} + c z_{3} = 0

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