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Position of a Point with Respect to Circle

If z1 , z...

Question

If $z_{1}, z_{2}, z_{3}, z_{4}$ be the vertices of a square in Argand plane, then prove that
$2 z_{2} = (1 + i) z_{1} + (1 - i) z_{3}$
and $2 z_{4} = (1 - i) z_{1} + (1 + i) z_{3}$

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Solution

We want the values of $z_{2}$ and $z_{4}$ in terms of $z_{1}$ and $z_{3}$
Hence we rotate about B and D.
Rotation about B in anti - clockwise sense
$z_{1} - z_{2} = (z_{3} - z_{2}) e^{π i / 2} = i (z_{3} - z_{2})$
$z_{1} - i z_{3} = z_{2} (1 - i) = \frac{z_{2} - 2}{1 + i}$
$2 z_{2} = (1 + i) z_{1} - i (1 + i) z_{3}$
or $2 z_{2} = (1 + i) z_{1} + (1 - i) z_{3}$
Similarly rotation about D will give $z_{4}$ in terms of $z_{1}$ . and $z_{3}$

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Similar questions

z_{1}, z_{2}

z_{3}

A, B

C

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

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

△ A B C

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

(d) ABC form an obtuse angle triangle

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

C (z_{2})

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

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

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

(d) ABC form an obtuse angled triangle

z_{1}, z_{2}, z_{3}, z_{4}

∠ C B A = π / 3

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

2 z_{4} = z_{1} (1 - i \sqrt{3}) + z_{3} (1 + i \sqrt{3})

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