If x + iy = √((a+ib)/(c+id)), prove...

MathematicsConjugate and ModulusJEE Advanced 1979Easy

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

Visualized Solution (English)

Given Equation

Given equation: $x + i y = \frac{a + ib}{c + i d}$
Objective: Prove $(x^{2} + y^{2})^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$

The Modulus Tool

Definition of Modulus: For $z = x + i y$ , $∣ z ∣ = x^{2} + y^{2}$
Strategy: Apply the modulus operator to both sides of the given equation.

Applying Modulus to Both Sides

Taking modulus: $∣ x + i y ∣ = \frac{a + ib}{c + i d}$

Using Modulus Properties

Property 1: $∣ z ∣ = ∣ z ∣$
Property 2: $\frac{z _{1}}{z _{2}} = \frac{∣ z _{1} ∣}{∣ z _{2} ∣}$
Applying properties: $∣ x + i y ∣ = \frac{∣ a + ib ∣}{∣ c + i d ∣}$

Substitution and First Squaring

Substitute values: $x^{2} + y^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$
Squaring both sides: $x^{2} + y^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$

Final Squaring and Conclusion

Squaring again: $(x^{2} + y^{2})^{2} = (\frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}})^{2}$
Final Result: $(x^{2} + y^{2})^{2} = \frac{a ^{2} + b ^{2}}{c ^{2} + d ^{2}}$
Key Takeaway: Modulus properties are powerful for eliminating imaginary parts in complex identities.

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Conceptually Similar Problems

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(A)

-2

(B)

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(C)

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Let $ω = e^{iπ /3}$ , and $a, b, c, x, y, z$ be non-zero complex numbers such that $a + b + c = x, a + bω + c ω^{2} = y, a + b ω^{2} + c ω = z$ . Then the value of $\frac{∣ x ∣ ^{2} + ∣ y ∣ ^{2} + ∣ z ∣ ^{2}}{∣ a ∣ ^{2} + ∣ b ∣ ^{2} + ∣ c ∣ ^{2}}$ is .........

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

Visualized Solution (English)

Given Equation

The Modulus Tool

Applying Modulus to Both Sides

Using Modulus Properties

Substitution and First Squaring

Final Squaring and Conclusion

Conceptually Similar Problems

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

If $z = x - i y$ and $z^{1/3} = p + i q$ , then $(x / p + y / q) / (p^{2} + q^{2})$ is equal to

Let $ω = e^{iπ /3}$ , and $a, b, c, x, y, z$ be non-zero complex numbers such that $a + b + c = x, a + bω + c ω^{2} = y, a + b ω^{2} + c ω = z$ . Then the value of $\frac{∣ x ∣ ^{2} + ∣ y ∣ ^{2} + ∣ z ∣ ^{2}}{∣ a ∣ ^{2} + ∣ b ∣ ^{2} + ∣ c ∣ ^{2}}$ is .........

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

Let the complex number $z_{1}, z_{2}, z_{3}$ be the vertices of an equilateral triangle. Let $z_{0}$ be the circumcentre of the triangle. Then prove that $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}$ .

Express $\frac{1}{1 - cos θ + 2 i sin θ}$ in the form $x + i y$ .

Prove that $cos tan^{- 1} sin cot^{- 1} x = \frac{x ^{2} + 1}{x ^{2} + 2}$ .

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

Given $α + β - γ = π$ , prove that $sin^{2} α + sin^{2} β - sin^{2} γ = 2 sin α sin β cos γ$ .

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

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

If $z = x - i y$ and $z^{1/3} = p + i q$ , then $(x / p + y / q) / (p^{2} + q^{2})$ is equal to

Let $ω = e^{iπ /3}$ , and $a, b, c, x, y, z$ be non-zero complex numbers such that $a + b + c = x, a + bω + c ω^{2} = y, a + b ω^{2} + c ω = z$ . Then the value of $\frac{∣ x ∣ ^{2} + ∣ y ∣ ^{2} + ∣ z ∣ ^{2}}{∣ a ∣ ^{2} + ∣ b ∣ ^{2} + ∣ c ∣ ^{2}}$ is .........

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

Let the complex number $z_{1}, z_{2}, z_{3}$ be the vertices of an equilateral triangle. Let $z_{0}$ be the circumcentre of the triangle. Then prove that $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}$ .

Express $\frac{1}{1 - cos θ + 2 i sin θ}$ in the form $x + i y$ .

Prove that $cos tan^{- 1} sin cot^{- 1} x = \frac{x ^{2} + 1}{x ^{2} + 2}$ .

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

Given $α + β - γ = π$ , prove that $sin^{2} α + sin^{2} β - sin^{2} γ = 2 sin α sin β cos γ$ .

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x+iy=c+ida+ib​​, prove that (x2+y2)2=c2+d2a2+b2​.

Visualized Solution (English)

Given Equation

The Modulus Tool

Applying Modulus to Both Sides

Using Modulus Properties

Substitution and First Squaring

Final Squaring and Conclusion

Conceptually Similar Problems

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If iz3+z2−z+i=0, then show that ∣z∣=1.

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If iz3+z2−z+i=0, then show that ∣z∣=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=x−iy and z1/3=p+iq, then (x/p+y/q)/(p2+q2) is equal to

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given α+β−γ=π, prove that sin2α+sin2β−sin2γ=2sinαsinβcosγ.

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

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

If $z = x - i y$ and $z^{1/3} = p + i q$ , then $(x / p + y / q) / (p^{2} + q^{2})$ is equal to

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

Express $\frac{1}{1 - cos θ + 2 i sin θ}$ in the form $x + i y$ .

Prove that $cos tan^{- 1} sin cot^{- 1} x = \frac{x ^{2} + 1}{x ^{2} + 2}$ .

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

Given $α + β - γ = π$ , prove that $sin^{2} α + sin^{2} β - sin^{2} γ = 2 sin α sin β cos γ$ .

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

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

If $z = x - i y$ and $z^{1/3} = p + i q$ , then $(x / p + y / q) / (p^{2} + q^{2})$ is equal to

For complex numbers $z$ and $w$ , prove that $∣ z ∣^{2} w - ∣ w ∣^{2} z = z - w$ if and only if $z = w$ or $z \overset{w}{ˉ} = 1$ .

Express $\frac{1}{1 - cos θ + 2 i sin θ}$ in the form $x + i y$ .

Prove that $cos tan^{- 1} sin cot^{- 1} x = \frac{x ^{2} + 1}{x ^{2} + 2}$ .

In any triangle $A B C$ , prove that $cot \frac{A}{2} + cot \frac{B}{2} + cot \frac{C}{2} = cot \frac{A}{2} cot \frac{B}{2} cot \frac{C}{2}$ .

Given $α + β - γ = π$ , prove that $sin^{2} α + sin^{2} β - sin^{2} γ = 2 sin α sin β cos γ$ .