If x-i y3-u-i v- then show that u-x-u-y-4x2-y2

(x+iy)^3=u+iv ⇒ x^3+i^3y^3+3x^2yi+3xy^2i^2=u+iv ⇒ (x^3 3xy^2)+(3x^2y y^3)i=u+iv Comparing both sides u=x(x^2 3y^2) and v=y(3x^2 y^2) u/x+u/y=x(x^2 3y^2)/x+y(3x^ ...

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Byju's Answer

Standard X

Mathematics

Quadratic Equations

If x+i y3=u+i...

Question

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

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Solution

$(x + i y)^{3} = u + i v$

$\Rightarrow x^{3} + i^{3} y^{3} + 3 x^{2} y i + 3 x y^{2} i^{2} = u + i v$

$\Rightarrow (x^{3} - 3 x y^{2}) + (3 x^{2} y - y^{3}) i = u + i v$

Comparing both sides

$u = x (x^{2} - 3 y^{2}) a n d v = y (3 x^{2} - y^{2})$

$\frac{u}{x} + \frac{u}{y} = \frac{x (x^{2} - 3 y^{2})}{x} + \frac{y (3 x^{2} - y^{2})}{y}$

$= x^{2} - 3 y^{2} + 3 x^{2} - y^{2}$

$= 4 x^{2} - 4 y^{2} = 4 (x^{2} - y^{2})$

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Similar questions

If (x + iy)³ = u + iv, then show that.

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\frac{u}{x} + \frac{v}{y} = 4 (x^{2} - y^{2})

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\frac{u}{x} + \frac{v}{y} =

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If (x+iy)3=u+iv, then show that ux+uy=4(x2−y2)

(x+iy)3=u+iv ⇒ x3+i3y3+3x2yi+3xy2i2=u+iv ⇒ (x3−3xy2)+(3x2y−y3)i=u+iv Comparing both sides u=x(x2−3y2) and v=y(3x2−y2) ux+uy=x(x2−3y2)x+y(3x2−y2)y =x2−3y2+3x2−y2 =4x2−4y2=4(x2−y2)

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