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Question: If \(\frac{(a + i)^{2}}{2a - 1}\) = p + iq, then p<sup>2</sup> + q<sup>2</sup> =...

If (a+i)22a−1\frac{(a + i)^{2}}{2a - 1} = p + iq, then p2 + q2 =

A

(a2+1)22a2+1\frac{(a^{2} + 1)^{2}}{2a^{2} + 1}

B

(a2+1)22a2−1\frac{(a^{2} + 1)^{2}}{2a^{2} - 1}

C

(a2+1)2(2a2−1)2\frac{(a^{2} + 1)^{2}}{(2a^{2} - 1)^{2}}

D

None of these

Answer

None of these

Explanation

Solution

Sol. p + iq = (a+i)22a−1\frac{(a + i)^{2}}{2a - 1} ………(1)

changing i to – i, then

p – iq = (a−i)2(2a−1)\frac{(a - i)^{2}}{(2a - 1)} …….(2)

multiplicative (1) and (2) (p + iq) . (p – iq) =((a+i)(a−i))2(2a−1).(2a+1)\frac{((a + i)(a - i))^{2}}{(2a - 1).(2a + 1)}

p2 + q2 = (a2+1)24a2−1\frac{(a^{2} + 1)^{2}}{4a^{2} - 1}