### Key Equations

Horizontal ellipse, center at origin | $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\phantom{\rule{0.8em}{0ex}}\text{}a>b$ |

Vertical ellipse, center at origin | $\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1,\phantom{\rule{0.8em}{0ex}}\text{}a>b$ |

Horizontal ellipse, center $(h,k)$ | $\frac{{\left(x\xe2\u02c6\u2019h\right)}^{2}}{{a}^{2}}+\frac{{\left(y\xe2\u02c6\u2019k\right)}^{2}}{{b}^{2}}=1,\phantom{\rule{0.8em}{0ex}}\text{}a>b$ |

Vertical ellipse, center $(h,k)$ | $\frac{{\left(x\xe2\u02c6\u2019h\right)}^{2}}{{b}^{2}}+\frac{{\left(y\xe2\u02c6\u2019k\right)}^{2}}{{a}^{2}}=1,\phantom{\rule{0.8em}{0ex}}\text{}a>b$ |

Hyperbola, center at origin, transverse axis on x-axis |
$\frac{{x}^{2}}{{a}^{2}}\xe2\u02c6\u2019\frac{{y}^{2}}{{b}^{2}}=1$ |

Hyperbola, center at origin, transverse axis on y-axis |
$\frac{{y}^{2}}{{a}^{2}}\xe2\u02c6\u2019\frac{{x}^{2}}{{b}^{2}}=1$ |

Hyperbola, center at $(h,k),$ transverse axis parallel to x-axis |
$\frac{{\left(x\xe2\u02c6\u2019h\right)}^{2}}{{a}^{2}}\xe2\u02c6\u2019\frac{{\left(y\xe2\u02c6\u2019k\right)}^{2}}{{b}^{2}}=1$ |

Hyperbola, center at $(h,k),$ transverse axis parallel to y-axis |
$\frac{{\left(y\xe2\u02c6\u2019k\right)}^{2}}{{a}^{2}}\xe2\u02c6\u2019\frac{{\left(x\xe2\u02c6\u2019h\right)}^{2}}{{b}^{2}}=1$ |

Parabola, vertex at origin, axis of symmetry on x-axis |
${y}^{2}=4px$ |

Parabola, vertex at origin, axis of symmetry on y-axis |
${x}^{2}=4py$ |

Parabola, vertex at $(h,k),$ axis of symmetry on x-axis |
${\left(y\xe2\u02c6\u2019k\right)}^{2}=4p\left(x\xe2\u02c6\u2019h\right)$ |

Parabola, vertex at $(h,k),$ axis of symmetry on y-axis |
${\left(x\xe2\u02c6\u2019h\right)}^{2}=4p\left(y\xe2\u02c6\u2019k\right)$ |

General Form equation of a conic section | $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ |

Rotation of a conic section | $$\begin{array}{l}x={x}^{\xe2\u20ac\xb2}\mathrm{cos}\phantom{\rule{0.8em}{0ex}}\text{}\mathrm{\xce\xb8}\xe2\u02c6\u2019{y}^{\xe2\u20ac\xb2}\mathrm{sin}\phantom{\rule{0.8em}{0ex}}\text{}\mathrm{\xce\xb8}\hfill \\ y={x}^{\xe2\u20ac\xb2}\mathrm{sin}\phantom{\rule{0.8em}{0ex}}\text{}\mathrm{\xce\xb8}+{y}^{\xe2\u20ac\xb2}\mathrm{cos}\phantom{\rule{0.8em}{0ex}}\text{}\mathrm{\xce\xb8}\hfill \end{array}$$ |

Angle of rotation | $\mathrm{\xce\xb8},\text{where}\mathrm{cot}\left(2\mathrm{\xce\xb8}\right)=\frac{A\xe2\u02c6\u2019C}{B}$ |