\(\dfrac{bx^4+ay^4}{ab}=\dfrac{1}{a+b}\Leftrightarrow\left(a+b\right)\left(bx^4+ay^4\right)=ab=ab.1^2=ab\left(x^2+y^2\right)^2\)
\(\Leftrightarrow\left(ab+b^2\right)x^4+\left(ab+a^2\right)y^4=ab\left(x^2+y^2\right)^2\)
\(\Leftrightarrow ab\left(x^4+y^4\right)+b^2x^4+a^2y^4=ab\left(x^4+y^4\right)+2abx^2y^2\)
\(\Leftrightarrow b^2x^4+a^2y^4-2abx^2y^2=0\)
\(\Leftrightarrow\left(bx^2-ay^2\right)^2=0\)
\(\Leftrightarrow bx^2=ay^2\Leftrightarrow\dfrac{x^2}{a}=\dfrac{y^2}{b}=\dfrac{x^2+y^2}{a+b}=\dfrac{1}{a+b}\)
\(\Leftrightarrow\dfrac{x^{200}}{a^{100}}=\dfrac{y^{200}}{b^{100}}=\dfrac{1}{\left(a+b\right)^{100}}\)
\(\Rightarrow\dfrac{x^{200}}{a^{100}}+\dfrac{y^{200}}{b^{100}}=\dfrac{2}{\left(a+b\right)^{100}}\)
