Question

Cho \(x;y\in R\) thỏa mãn \(x\sqrt{1-y^2}+y\sqrt{1-x^2}=1\). Tính A = \(5x^2+5y^2\)

Answer 1

Lời giải:

Ta có:

\(x\sqrt{1-y^2}+y\sqrt{1-x^2}=1\)

\(\Leftrightarrow x\sqrt{1-y^2}=1-y\sqrt{1-x^2}\)

\(\Rightarrow x^2(1-y^2)=1+y^2(1-x^2)-2y\sqrt{1-x^2}\) (bình phương hai vế)

\(\Leftrightarrow x^2=1+y^2-2y\sqrt{1-x^2}\)

\(\Leftrightarrow y^2+(1-x^2)-2y\sqrt{1-x^2}=0\)

\(\Leftrightarrow (y-\sqrt{1-x^2})^2=0\)

\(\Rightarrow y-\sqrt{1-x^2}=0\Rightarrow y=\sqrt{1-x^2}\)

\(\Rightarrow y^2=1-x^2\Leftrightarrow x^2+y^2=1\)

Do đó: \(A=5x^2+5y^2=5(x^2+y^2)=5\)