Question 136

If $$x^4 + x^2y^2 + y^4 = 133$$ and $$x^2 - xy + y^2 = 7$$,then what is the value of xy?

Solution

As per the given question,

$$x^4 + x^2y^2 + y^4 = 133--------(i)$$

$$x^2 - xy + y^2 = 7----(ii)$$

Taking the square of the equation (ii)

$$\Rightarrow (x^2 - xy + y^2)^2 = 7^2$$

$$\Rightarrow x^4+x^2y^2+y^4+2(-x^3y-y^3x+x^2y^2)=49$$

$$\Rightarrow 133-2xy(x^2-xy+y^2)=49$$

$$\Rightarrow 133-2xy\times 7=49$$

$$\Rightarrow 2xy\times 7=133-49$$

$$\Rightarrow 14xy=84$$

$$\Rightarrow xy=6$$

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