If x - iy = √(a - ib)/(c - id) , Prove that: (x² + y²)² = (a²+ b²)/(c² + d²)

Solution:

It is given that,

x - iy = √(a - ib)/(c - id)

We will rationalize the denominator here

x - iy = √[ (a - ib)/(c - id) · (c + id) / (c + id) ]

= √ [(ac + bd) + i (ad - bc)] / [c² + d²]

Squaring on both sides,

(x² - y²) - i (2xy) = (ac + bd) / (c² + d²) + i (ad - bc) / (c² + d²)

Comparing the real and imaginary parts,

x² - y² = (ac + bd) / (c² + d²) and 2xy = (ad - bc) / (c² + d²) ... (1)

Now we will start with LHS of what needs to be proved.

LHS = (x² + y²)²

= (x² - y²)² + 4x²y²

= (ac + bd)² / (c² + d²)² + (ad - bc)² / (c² + d²)² (From (1))

= (a²c² + b²d² + 2abcd) / (c² + d²)² + (a²d² + b²c^{2 -} 2abcd) / (c² + d²)²

= (a²c² + b²d² + a²d² + b²c² ​​​​) / (c² + d²)²

= (a² (c² + d²) + b² (c² + d²)) / (c² + d²)²

= (c² + d²)​ (a² + b²) / (c² + d²)²

= (a² + b²) / (c² + d²)

= RHS

Hence we proved that (x² + y²)² = (a²+ b²)/(c² + d²)

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NCERT Solutions Class 11 Maths Chapter 5 Exercise ME Question 4

If x - iy = √(a - ib)/(c - id) , Prove that: (x² + y²)² = (a²+ b²)/(c² + d²)

Solution:

Hence we proved that (x² + y²)² = (a²+ b²)/(c² + d²)