Complex_01

Algebra Level pending

If x + y i x+yi , a + b i a+bi and a − b i a-bi are 3 complex numbers verifying the relation x + y i = a + b i a − b i x+yi=\frac{a + bi}{a - bi} , then what is what is to value of x 2 + y 2 x^{2}+y^{2} ?

N.B. x , y , a , b x,y,a,b are real numbers and i i = − 1 \sqrt{-1}

a 2 − b 2 a^{2}-b^{2} a 2 + b 2 a^{2}+b^{2} 1 1 2 a b 2ab

Achraf Wassim by Achraf Wassim , 41, Egypt

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1 solution

x + i y = a + i b a − i b = a 2 − b 2 + 2 i a b a 2 + b 2 x+iy=\dfrac {a+ib}{a-ib}=\dfrac {a^2-b^2+2iab}{a^2+b^2}

⟹ x 2 + y 2 = ( a 2 − b 2 ) 2 + 4 a 2 b 2 ( a 2 + b 2 ) 2 \implies x^2+y^2=\dfrac {(a^2-b^2)^2+4a^2b^2}{(a^2+b^2)^2}

= ( a 2 + b 2 ) 2 ( a 2 + b 2 ) 2 = 1 =\dfrac {(a^2+b^2)^2}{(a^2+b^2)^2}=\boxed 1 .

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