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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)
= (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
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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