## 2016年11月29日火曜日

### 数学 - 2次曲線と直線 - 2次曲線 – 2次曲線と直線(2次曲線の回転)

1. $\begin{array}{l}x\mathrm{cos}{45}^{°}+y\mathrm{sin}{45}^{°}=\frac{x+y}{\sqrt{2}}\\ -x\mathrm{sin}{45}^{°}+y\mathrm{cos}{45}^{°}=\frac{-x+y}{\sqrt{2}}\\ 3·\frac{{\left(x+y\right)}^{2}}{2}+2\frac{{x}^{2}-{y}^{2}}{2}+3\frac{{\left(x-y\right)}^{2}}{2}=2\\ 8{x}^{2}+4{y}^{2}=4\\ 2{x}^{2}+{y}^{2}=1\\ 楕円\end{array}$

2. $\begin{array}{l}x\mathrm{cos}{30}^{°}+y\mathrm{sin}{30}^{°}=\frac{x+\sqrt{3}y}{2}\\ -x\mathrm{sin}{30}^{°}+y\mathrm{cos}{30}^{°}=\frac{-\sqrt{3}x+y}{2}\\ 2\frac{{x}^{2}+2\sqrt{3}xy+3{y}^{2}}{4}+2\sqrt{3}\frac{-\sqrt{3}{x}^{2}+\sqrt{3}{y}^{2}-2xy}{4}=-1\\ \frac{-4{x}^{2}+12{y}^{2}}{4}=-1\\ {x}^{2}-3{y}^{2}=1\\ 双曲線\end{array}$

3. $\begin{array}{l}x\mathrm{cos}{45}^{°}+y\mathrm{sin}{45}^{°}=\frac{x+y}{\sqrt{2}}\\ -x\mathrm{sin}{45}^{°}+y\mathrm{cos}{45}^{°}=\frac{-x+y}{\sqrt{2}}\\ 2{y}^{2}=2x\\ {y}^{2}=x\\ 放物線\end{array}$

4. $\begin{array}{l}x\mathrm{cos}-{45}^{°}+y\mathrm{sin}-{45}^{°}=\frac{x-y}{\sqrt{2}}\\ -x\mathrm{sin}-{45}^{°}+y\mathrm{cos}-{45}^{°}=\frac{x+y}{\sqrt{2}}\\ \frac{{\left(x-y\right)}^{2}}{2}-\frac{{x}^{2}-{y}^{2}}{2}+\frac{{\left(x+y\right)}^{2}}{2}=3\\ {x}^{2}+2{y}^{2}=6\\ \frac{{x}^{2}}{6}+\frac{{y}^{2}}{2}=1\\ 楕円\end{array}$

5. $\begin{array}{l}x\mathrm{cos}{60}^{°}+y\mathrm{sin}{60}^{°}=\frac{x+\sqrt{3}y}{2}\\ -x\mathrm{sin}{60}^{°}+y\mathrm{cos}{60}^{°}=\frac{-\sqrt{3}x+y}{2}\\ {\left(\frac{\sqrt{3}x+y}{2}+\frac{-\sqrt{3}x+y}{2}\right)}^{2}=4\left(\frac{x+\sqrt{3}y}{2}-\frac{-x+\sqrt{3}y}{2}\right)\\ {y}^{2}=4x\\ 放物線\end{array}$

6. $\begin{array}{l}x\mathrm{cos}{45}^{°}+y\mathrm{sin}{45}^{°}=\frac{x+y}{\sqrt{2}}\\ -x\mathrm{sin}{45}^{°}+y\mathrm{cos}{45}^{°}=\frac{-x+y}{\sqrt{2}}\\ \frac{{\left(x+y\right)}^{2}}{2}-6\frac{{x}^{2}-{y}^{2}}{2}+\frac{{\left(x-y\right)}^{2}}{2}=-4\\ -4{x}^{2}+8{y}^{2}=-8\\ \frac{{x}^{2}}{2}-{y}^{2}=1\\ 双曲線\end{array}$