If x - yi 3- u - vi- prove that u - x - v - y -4 x 2- y 2

(x+yi)^3=u+vi. ⇒ x^3+3x^2yi+3xy^2i^2+y^3i^3=u+vi ⇒ x^3+3x^2yi 3xy^2 y^3i=u+vi ⇒ x^3 3xy^2+(3x^2y y^3)i=u+vi Equating real and imaginary parts, we have x^3 3xy^2 ...

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Standard XII

Mathematics

Complex Numbers

If x + yi 3= ...

Question

If $(x + y i)^{3} = u + v i,$ prove that $\frac{u}{x} + \frac{v}{y} = 4 (x^{2} - y^{2}) .$

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Solution

$(x + y i)^{3} = u + v i . \Rightarrow x^{3} + 3 x^{2} y i + 3 x y^{2} i^{2} + y^{3} i^{3} = u + v i \Rightarrow x^{3} + 3 x^{2} y i - 3 x y^{2} - y^{3} i = u + v i \Rightarrow x^{3} - 3 x y^{2} + (3 x^{2} y - y^{3}) i = u + v i$
Equating real and imaginary parts, we have
$x^{3} - 3 x y^{2} = u, 3 x^{2} y - y^{3} = v \Rightarrow \frac{u}{x} = x^{2} - 3 y^{2}, \frac{v}{y} = 3 x^{2} - y^{2} \Rightarrow \frac{u}{x} + \frac{v}{y} = x^{2} - 3 y^{2} + 3 x^{2} - y^{2} = 4 x^{2} = 4 (x^{2} - y^{2})$

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If (x+yi)3=u+vi, prove that ux+vy=4(x2−y2).

(x+yi)3=u+vi.⇒x3+3x2yi+3xy2i2+y3i3=u+vi⇒x3+3x2yi−3xy2−y3i=u+vi⇒x3−3xy2+(3x2y−y3)i=u+vi Equating real and imaginary parts, we have x3−3xy2=u,3x2y−y3=v⇒ux=x2−3y2,vy=3x2−y2⇒ux+vy=x2−3y2+3x2−y2=4x2=4(x2−y2)

If $(x + y i)^{3} = u + v i,$ prove that $\frac{u}{x} + \frac{v}{y} = 4 (x^{2} - y^{2}) .$