Given-x-4-x-2-y-2-y-4-133-and-x-2-xy-y-2-7-then-what-is-the-value-of-xy-

Question Number 109577 by bemath last updated on 24/Aug/20

G i v e n x^{4} + x^{2} y^{2} + y^{4} = 133

a n d x^{2} - x y + y^{2} = 7

t h e n w h a t i s t h e v a l u e o f x y ?

Commented by bemath last updated on 24/Aug/20

c o o l l \dots . .

Answered by Dwaipayan Shikari last updated on 24/Aug/20

{(x^{2} + y^{2})}^{2} - x^{2} y^{2} = 133

{(7 + x y)}^{2} - x^{2} y^{2} = 133

49 + 14 x y + x^{2} y^{2} - x^{2} y^{2} = 133

14 x y = 84

x y = 6

Answered by john santu last updated on 24/Aug/20

\frac{⇋ J S ⇌}{♠}

(1) x^{4} + 2 x^{2} y^{2} + y^{4} - x^{2} y^{2} = 133

\Rightarrow {(x^{2} + y^{2})}^{2} - {(x y)}^{2} = 133

\Rightarrow (x^{2} - x y + y^{2}) (x^{2} + x y + y^{2}) = 133

(2) x^{2} - x y + y^{2} = 7

\Rightarrow 7 (x^{2} - x y + y^{2} + 2 x y) = 133

\Rightarrow 7 (7 + 2 x y) = 133

\Rightarrow 7 + 2 x y = 19 \Rightarrow 2 x y = 12

∴ x y = 6

Commented by mnjuly1970 last updated on 24/Aug/20

Answered by Rasheed.Sindhi last updated on 24/Aug/20

\frac{\frac{(x^{2} - y^{2}) (x^{4} + x^{2} y^{2} + y^{4})}{x^{2} - y^{2}}}{\frac{(x + y) (x^{2} - x y + y^{2})}{x + y}} = \frac{133}{7} = 19

\frac{x^{6} - y^{6}}{x^{2} - y^{2}} \times \frac{x + y}{x^{3} + y^{3}} = 19

\frac{(x^{3} - y^{3}) (x^{3} + y^{3})}{(x - y) (x^{3} + y^{3})} = 19

\frac{(x - y) (x^{2} + x y + y^{2})}{(x - y)} = 19

x^{2} + x y + y^{2} = 19 \dots \dots . . (i)

x^{2} - x y + y^{2} = 7 \dots \dots \dots (i i)

(i) - (i i) : 2 x y = 12

x y = 6

Answered by Rasheed.Sindhi last updated on 24/Aug/20

x^{4} + x^{2} y^{2} + y^{4} = 133 \dots \dots \dots (i)

x^{2} - x y + y^{2} = 7 \dots \dots \dots \dots . . (i i)

\frac{x^{4} + x^{2} y^{2} + y^{4}}{x^{2} - x y + y^{2}} = \frac{133}{7} = 19

\frac{(x^{2} - x y + y^{2}) (x^{2} + x y + y^{2})}{x^{2} - x y + y^{2}} = 19

x^{2} + x y + y^{2} = 19 \dots \dots \dots \dots . . (i i i)

(i i i) - (i i) : 2 x y = 12

x y = 6