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NCERT Class XI Mathematics - Complex Numbers and Quadratic Equations - Solutions

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Question : 43 of 52
Marks: +1, -0
If a + ib = (x+i)22x2+1\frac{(x+i)^2}{2x^2+1} , prove that a2+b2a^2+b^2 = (x2+1)2(2x2+1)2\frac{(x^2+1)^2}{(2x^2+1)^2}
Solution:  
We have, a + ib = (x+i)22x2+1\frac{(x+i)^2}{2x^2+1}
∴ |a + ib| = ∣(x+i)22x2+1∣\left|\frac{(x+i)^2}{2x^2+1}\right| = ∣(x+i)2∣∣2x2+1∣\frac{\left|(x+i)^2\right|}{\left|2x^2+1\right|} = ∣x+i∣2∣2x2+1∣\frac{\left|x+i\right|^2}{\left|2x^2+1\right|}
⇒ a2+b2\sqrt{a^2+b^2} = (x2+12)2(2x2+1)2\frac{\left(\sqrt{x^2+1^2}\right)^2}{\left(\sqrt{2x^2+1}\right)^2} ... (i)
Squaring (i) both sides, we get
a2+b2a^2+b^2 = (x2+1)2(2x2+1)2\frac{(x^2+1)^2}{(2x^2+1)^2}
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