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1.
Equation of an Ellipse
$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$
2.
Equation of a Hyperbola
$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
3.
Parametric equation of the right branch
$\begin{array}{ll} x = a \cosh t \\[1 em] y=b \sinh t \end{array} 0\le t \le 2\pi$