Question

(a)
8.If a4 + a2b2 +b4 = 8 and a2 + ab + b2 = 4, then the value of ab is
यदि a4 + a2b2 + b4 = 8 और a2 + ab + b2 = 4, तो ab का मान है।
(a) -1 (b) 0
(c) 2
1
43.13+03-3abc
(d)​

Answer 1

$\begin{gathered}\begin{gathered}\bf\: Given-\begin{cases} &\sf{ {a}^{4}+{a}^{2} {b}^{2}+{b}^{4} = 8} \\ &\sf{ {a}^{2} + ab + {b}^{2} = 4} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To \: Find - \begin{cases} &\sf{value \: of \: ab}\end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{{\underline{Formula \: Used - }}}}  \end{gathered}$

$\boxed{\red{\sf\: {x}^{2} - {y}^{2} + (x + y)(x - y)}}$

$\boxed{\red{\sf\: {x}^{2} + 2xy + {y}^{2} = {(x + y)}^{2}}}$

$\large\underline{\sf{Solution-}}$

Given that,

$\rm :\longmapsto\: {a}^{2} + ab + {b}^{2} = 4 - - - (1)$

and

$\rm :\longmapsto\: {a}^{4} + {a}^{2} {b}^{2} + {b}^{4} = 8$

$\rm :\longmapsto\: {a}^{4} + {a}^{2} {b}^{2} + {b}^{4} + {a}^{2} {b}^{2} - {a}^{2} {b}^{2}= 8$

$\rm :\longmapsto\: {a}^{4}+ {b}^{4} + 2{a}^{2} {b}^{2} - {a}^{2} {b}^{2}= 8$

$\rm :\longmapsto\: {\bigg( {a}^{2} + {b}^{2} \bigg) }^{2} - {(ab)}^{2} = 8$

$\rm :\longmapsto\:( {a}^{2} + {b}^{2} + ab)( {a}^{2} + {b}^{2} - ab) = 8$

$\rm :\longmapsto\:4( {a}^{2} + {b}^{2} - ab) = 8 \: \: \: \: \: \: \: \: \: \boxed{\red{\sf\:using \: (1)}}$

$\rm :\longmapsto\:{a}^{2} + {b}^{2} - ab=2$

$\rm :\longmapsto\:4 - ab - ab=2 \: \: \: \: \: \: \: \: \boxed{\red{\sf\:using \: (1)}}$

$\rm :\longmapsto\:4 - 2ab = 2$

$\rm :\longmapsto\:2ab = 4 - 2$

$\rm :\longmapsto\:2ab = 2$

$\bf\implies \:ab = 1$

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)