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Question

If a point in argand plane A(3,2) rotated through B(1,1) about π4 to obtain C, then area of ABC is

A
54 sq. unit
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B
52 sq. unit
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C
502 sq. unit
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D
504 sq. unit
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Solution

The correct option is D 504 sq. unit
Let zA=3+2i,zB=1+i and after rotating zA the complex number be zC then
|zAzB|=|zCzB|
Using rotation property we have
zCzBzAzB=|zCzB||zAzB|eiπ/4zCzB2+i=12+i2zC=(1+i)+(12+3i2)zC=1+22+(3+2)i2
Now AC=|zAzC|=12(221)2+(23)2
=1052 unit

Now BD=zB(zA+zC2) [AB=CB]
BD=10+522 unit
Hence required area =AC×BD2
=504 sq. unit
(Here we have considers anticlockwise direction however clock wise direction will also give same answer)

Alternte Solution:
Using sine rule
required area will be =12(AB)(BC)sin(π/4)
=122(AB)2=122[22+12]
=504 sq. unit

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