题目
已知 \(C_1:x^2+y^2=a\)，\(C_2:x^2+xy+y^2=x^4+x^3y+x^2y^2+xy^3+y^4\)，求 \(|C_1 \cap C_2|\)。(By \(\text{Geometry11}\))
解答(by \(\text{2021hych}\))
解：考虑一个线性变换 \(\rho :\mathbb{R}^2 \to \mathbb{R}^2\)，其中 \(\rho ((x,y))=(\frac{x+y}{2},\frac{x-y}{2})\)，则 \(\rho^{-1} ((x,y))=(x+y,x-y)\)，从而 \(\rho\) 是一个双射，记 \(C_3=\rho(C_1)\)，\(C_4=\rho(C_2)\)，则:
\[C_3:(x+y)^2+(x-y)^2=a \Rightarrow C_3:x^2+y^2=\frac{a}{2}
\]
\[C_4:(x+y)^2+(x+y)(x-y)+(x-y)^2=(x+y)^4+(x+y)^3(x-y)+(x+y)^2(x-y)^2+(x+y)(x-y)^3+(x-y)^4
\]
\[\Rightarrow C_4:3x^2+y^2=5x^4+10x^2y^2+y^4
\]