If x-iy-a-ibc-id prove that x2-y2-ac-bdc2-d2-

#160; x iy=√(a ibc id)=√((a ib)(c+id)c^2+d^2) x^2 y^2 2ixy=(a ib)(c+id)c^2+d^2 (squaring both sides) ⇒ x^2 y^2=ac+bdc^2+d^2 (Compare real pats) ...

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Properties of Conjugate of a Complex Number

If x-iy=√a-...

Question

If $x - i y = \sqrt{\frac{a - i b}{c - i d}}$ prove that $x^{2} - y^{2} = \frac{a c + b d}{c^{2} + d^{2}}$ .

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Solution

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x - i y = \frac{a - i b}{c - i d}

x^{2} + y^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}

(x + i y) = \sqrt{\frac{a + i b}{c + i d}}

(x^{2} + y^{2})^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}

\frac{(a + i b)}{(c + i d)} = x + i y

\frac{a - i b}{c - i d} = x - i y

\frac{a^{2} + b^{2}}{c^{2} + d^{2}} = x^{2} + y^{2}

x + i y = \sqrt{\frac{a + i b}{c + i d}}

(x^{2} + y^{2})^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}

If $x - i y = \sqrt{\frac{a - i d}{c - i d}},$ prove that $(x^{2} + y^{2})^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}$ .

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