Question

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Answer 1

Answer:

x² + y² = 1

Step-by-step explanation:

$Given:x + iy=\sqrt{\frac{1+i}{1-i}}$

On squaring both sides, we get

$=>(x + iy)^2=\frac{1+i}{1-i}$

On rationalizing, we get

$=>(x + iy)^2 =\frac{1+i}{1-i}*\frac{1+i}{1+i}$

$=>x^2 + i^2y^2 + 2ixy = \frac{1^2 + i^2 + 2i}{(1- i^2)}$

We know that i² = -1.

$=>x^2-y^2+2ixy=\frac{1-1+2i}{1+1}$

$=>x^2-y^2+2ixy=i$

$=>x^2 - y^2 + 2ixy=0 + i(1)$

Comparing real and imaginary parts in L.H.S and R.H.S

(i)

x² - y² = 0

(ii)

2xy = 1

xy = 1/2

Now,

We know that (x² + y²)² = (x² - y²)² + 4(xy)²

                                      = 0 + 4(1/2)²

                                      = 1.

∴ (x² + y²) = 1.

Hope it helps!

Answer 2

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