If x+iy=a+ib/a-ib, prove that x2+y2=1?

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  • rotchm
    Lv 7
    3 weeks ago

    x+iy=(a+ib)/(a-ib). Take the conjugate each side to get

    x - iy = (a-ib)/(a+ib). Now multiply the RHS's together & LHS's together to get

    (x+iy)(x-iy) = (a+ib)(a-ib) / ( (a-ib)(a+bi) ).  Expand & simplify to get

    x² + y² = (a² + b²)/(a² + b²). What does the RHS equal?

    Done!

  • I presume you mean (a + ib) / (a - ib) and x^2 + y^2 = 1

    x + iy = (a + ib) / (a - ib)

    x + iy = (a + ib)^2 / (a^2 - i^2 * b^2)

    x + iy = (a + ib)^2 / (a^2 + b^2)

    x + iy = (a^2 + 2abi - b^2) / (a^2 + b^2)

    x = (a^2 - b^2) / (a^2 + b^2)

    y = 2ab / (a^2 + b^2)

    x^2 + y^2 =>

    (a^2 - b^2)^2 / (a^2 + b^2)^2 + (2ab)^2 / (a^2 + b^2)^2 =>

    ((a^2 - b^2)^2 + (2ab)^2) / (a^2 + b^2)^2 =>

    (a^4 - 2a^2 * b^2 + b^4 + 4a^2 * b^2) / (a^2 + b^2)^2 =>

    (a^4 + 2a^2 * b^2 + b^4) / (a^2 + b^2)^2 =>

    (a^2 + b^2)^2 / (a^2 + b^2)^2 =>

    1

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