Câu hỏi của Phạm Đức

Question

Cho $\frac{x^4}{a}+\frac{y^4}{b}=\frac{1}{a+b}$

$x^2+y^2=1$

CMR $bx^2=ay^2$

Nguyễn Việt Lâm · Accepted Answer

$x^2+y^2=1\Rightarrow\left(x^2+y^2\right)^2=1\Rightarrow x^4+y^4+2x^2y^2=1$

$\Rightarrow\frac{1}{a+b}=\frac{x^4+y^4+2x^2y^2}{a+b}$

Ta có:

$\frac{x^4}{a}+\frac{y^4}{b}=\frac{x^4+y^4+2x^2y^2}{a+b}\Leftrightarrow\frac{bx^4+ay^4}{ab}=\frac{x^4+y^4+2x^2y^2}{a+b}$

$\Leftrightarrow\left(bx^4+ay^4\right)\left(a+b\right)=ab\left(x^4+y^4+2x^2y^2\right)$

$\Leftrightarrow abx^4+b^2x^4+a^2y^4+aby^4=abx^4+aby^4+2abx^2y^2$

$\Leftrightarrow\left(bx^2\right)^2+\left(ay^2\right)^2-2abx^2y^2=0$

$\Leftrightarrow\left(bx^2-ay^2\right)^2=0$

$\Leftrightarrow bx^2-ay^2=0$

$\Rightarrow bx^2=ay^2$Câu trả lời: $x^2+y^2=1\Rightarrow\left(x^2+y^2\right)^2=1\Rightarrow x^4+y^4+2x^2y^2=1$