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)

= (x² - 1 + 2xi) / (2x² + 1)

= (x² - 1) / (2x² + 1) + 2xi / (2x² + 1)

Comparing the real and imaginary parts,

a = (x² - 1) / (2x² + 1) and b = 2x / (2x² + 1).

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

LHS = a² + b²

= (x² - 1)² / (2x² + 1)² + 4x² / (2x² + 1)²

= (x⁴ + 1 - 2x² + 4x²) / (2x² + 1)²

= (x⁴ + 1 + 2x²) / (2x² + 1)²

= (x² + 1)²/(2x² + 1)²

= RHS

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

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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)