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Question

If (a+i)22ai=p+iq, where aR, then the value of p2+q2 is

A
(a21)24a2+1
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B
(a2+1)24a2+1
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C
(a21)22a2+1
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D
(a2+1)22a2+1
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Solution

The correct option is B (a2+1)24a2+1
Given : (a+i)22ai=p+iq
Taking conjugate on both sides, we get
¯¯¯¯¯¯¯¯¯¯¯¯¯¯p+iq=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯((a+i)2(2ai))piq=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(a+i)2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(2ai) (¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z1z2)=¯¯¯¯¯z1¯¯¯¯¯z2)piq=(ai)2(2a+i) (¯¯¯¯¯¯¯¯¯¯(zn)=(¯¯¯z)n)

Now, p2+q2
=(p+iq)×(piq)=(a+i)22ai×(ai)2(2a+i)=(a2+1)24a2+1

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