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NCERT Class XI Mathematics - Complex Numbers and Quadratic Equations - Solutions

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Question : 36 of 52
Marks: +1, -0
If x - iy = a−ibc−id\sqrt{\frac{a-ib}{c-id}} , prove that (x2+y2)2(x^2+y^2)^2 = a2+b2c2+d2\frac{a^2+b^2}{c^2+d^2}
Solution:  
We have given, x - iy = a−ibc−id\sqrt{\frac{a-ib}{c-id}} ... (i)
Squaring (i) both sides, we get
(x−iy)2(x-iy)^2 = a−ibc−id\frac{a-ib}{c-id} ⇒ ∣(x−iy)2∣\left| (x-iy)^2 \right| = ∣a−ibc−id∣\left| \frac{a-ib}{c-id} \right|
⇒ ∣x−iy∣2\left| x-iy \right|^2 = ∣a−ib∣∣c−id∣\frac{\left| a-ib \right|}{\left| c-id \right|} ⇒ (x2+y2)2\left( \sqrt{x^2+y^2} \right)^2 = a2+b2c2+d2\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}}
⇒ x2+y2x^2+y^2 = a2+b2c2+d2\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}} ... (ii)
Squaring (ii) sides, we get
(x2+y2)2(x^2+y^2)^2 = a2+b2c2+d2\frac{a^2+b^2}{c^2+d^2}
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