10.7.2: Key Equations

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Key Equations

Horizontal ellipse, center at origin	$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, a > b$
Vertical ellipse, center at origin	$\frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1, a > b$
Horizontal ellipse, center $(h, k)$	$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1, a > b$
Vertical ellipse, center $(h, k)$	$\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1, a > b$

Hyperbola, center at origin, transverse axis on x-axis	$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
Hyperbola, center at origin, transverse axis on y-axis	$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$
Hyperbola, center at $(h, k),$ transverse axis parallel to x-axis	$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$
Hyperbola, center at $(h, k),$ transverse axis parallel to y-axis	$\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1$

Parabola, vertex at origin, axis of symmetry on x-axis	$y^{2} = 4 p x$
Parabola, vertex at origin, axis of symmetry on y-axis	$x^{2} = 4 p y$
Parabola, vertex at $(h, k),$ axis of symmetry on x-axis	${(y - k)}^{2} = 4 p (x - h)$
Parabola, vertex at $(h, k),$ axis of symmetry on y-axis	${(x - h)}^{2} = 4 p (y - k)$

General Form equation of a conic section	$A x^{2} + B x y + C y^{2} + D x + E y + F = 0$
Rotation of a conic section	$\begin{array}{l} x = x^{'} \cos θ - y^{'} \sin θ \\ y = x^{'} \sin θ + y^{'} \cos θ \end{array}$
Angle of rotation	$θ, where \cot (2 θ) = \frac{A - C}{B}$

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Key Equations