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# If a + ib = (x + i)²/ (2x² + 1) , prove that a² + b² = (x² + 1)²/(2x² + 1)²

**Solution:**

The given complex number is,

a + ib = (x + i)²/ (2x² + 1)

= (x^{2} - 1 + 2xi) / (2x^{2} + 1)

= (x^{2} - 1) / (2x^{2} + 1) + 2xi / (2x^{2} + 1)

Comparing the real and imaginary parts,

a = (x^{2} - 1) / (2x^{2} + 1) and b = 2x / (2x^{2} + 1).

Now, we will consider the LHS of what needs to be proved.

LHS = a² + b²

= (x^{2} - 1)^{2} / (2x^{2} + 1)^{2} + 4x^{2} / (2x^{2} + 1)^{2}

= (x^{4} + 1 - 2x^{2} + 4x^{2}) / (2x^{2} + 1)^{2}

= (x^{4} + 1 + 2x^{2}) / (2x^{2} + 1)^{2}

= (x² + 1)²/(2x² + 1)²

= RHS

Thus, we proved that a² + b² = (x² + 1)²/(2x² + 1)²

NCERT Solutions Class 11 Maths Chapter 5 Exercise ME Question 11

## If a + ib = (x + i)²/ (2x² + 1) , prove that a² + b² = (x² + 1)²/(2x² + 1)²

**Summary:**

We have proved that a² + b² = (x² + 1)²/(2x² + 1)² when a + ib = (x + i)²/ (2x² + 1)

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