If x-iy-a-ibc-id prove that x2-y2-a2-b2c2-d2-

Given, x iy=a ibc id ⇒ x+iy=a+ibc+id (x iy)(x+iy)=a ibc id×a+ibc+id x^2 (iy)^2=a^2 (ib)^2c^2 (id)^2 ∴ x^2+y^2=a^2+b^2c^2+d^2 Hence prove ...

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Properties of Conjugate of a Complex Number

If x-iy=a-i...

Question

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

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Solution

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\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}}

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

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

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

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