Question

यदि $(x + \iota y)^3 = u + \iota v$, तो दर्शाइए की $\dfrac{u}{x} + \dfrac{v}{y} = 4 (x^2 - y^2)$.

Answer 1

Answer:

Step-by-step explanation:

प्रश्नानुसार  

$(x + \iota y)^3 = u + \iota v$

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

अतः    $u=x^{3}-3xy^{2}$

तथा    $v=3x^{2}y-y^{3}$

$L.H.S.=\frac{u}{x} +\frac{v}{y} \\\\=\frac{x^{3}-3xy^{2}}{x} +\frac{3x^{2}y-y^{3}}{y} \\\\=x^{2}-3y^{2}+3x^{2}-y^{2}\\\\=4x^{2}-4y^{2}\\\\=4(x^{2}-y^{2})\\\\=R.H.S.$