Question

If $u+iv=(x+iy{)}^3$ then $\frac{u}{x}+\frac{v}{y}=$

• A. $4(x^2+y^2)$
• B. $4(x^2-y^2)$
• C. $2(x^2+y^2)$
• D. $2(x^2-y^2)$

Answer 1

The correct option is B $4(x^2-y^2)$

$u+iv=(x+iy{)}^3$
$=x^3+3x^{2}iy+3x{i}^2{y}^2+i^3{y}^3$
$=(x^3-3x{y}^2)+i(3x^{2}y-y^3)$

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