If (a+i)22a−i=p+iq, where a∈R, then the value of p2+q2 is
A
(a2−1)24a2+1
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B
(a2+1)24a2+1
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C
(a2−1)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)22a−i=p+iq
Taking conjugate on both sides, we get ¯¯¯¯¯¯¯¯¯¯¯¯¯¯p+iq=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯((a+i)2(2a−i))⇒p−iq=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(a+i)2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(2a−i)(∵¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z1z2)=¯¯¯¯¯z1¯¯¯¯¯z2)⇒p−iq=(a−i)2(2a+i)(∵¯¯¯¯¯¯¯¯¯¯(zn)=(¯¯¯z)n)