Ellipse, Parabola, Hyperbola

Ellipse with center $C(x_0 \textrm{ , } y_0)$ and major axis parallel to $x$ axis

Hyperbola
Length of major axis $A'A = 2a$
Length of minor axis $B'B = 2b$
Distance from center $C$ to focus $F$ or $F'$ is
$c = \sqrt{a^2 - b^2}$
Eccentricity = $\epsilon = \frac{c}{a} = \frac{\sqrt{a^2 - b^2}}{a}$
Equation in rectangular coordinates:
$\frac{(x - x_0)^2}{a^2} + \frac{(y - y_0)^2}{b^2} = 1$
Equation in polar coordinates if $C$ is at $O$:
$r^2 = \frac{a^2b^2}{a^2 \textrm{ sin }^2 \theta + b^2 \textrm{ cos }^2 \theta}$
Equation in polar coordinates if $C$ is on $x$ axis and $F'$ is at $O$:
$r = \frac{a(1 - c^2)}{1 - c \textrm{ cos } \theta}$
If $P$ is any point on the ellipse, $PF + PF' = 2a$
If the major axis is parallel to the $y$ axis, interchange $x$ and $y$ in the above or replace $\theta$ by $\frac{1}{2}\pi - \theta$ [or $90^\circ - \theta$]

Parabola with axis parallel to $x$ axis

If vertex is at $A(x_0 \textrm{ , } y_0)$ and the distance from $A$ to focus$f$ is $a > 0$, the equation of the parabola is if parabola opens to right

$(y - y_0)^2 = 4a(x - x_0)$
Parabola

If parabola opens to left
$(y - y_0)^2 = -4a(x - x_0)$
If parabola opens to left

If focus is at the origin the equation in polar coordinates is
$r = \frac{2a}{1 - \textrm{ cos } \theta}$

Parabola If parabola opens to left

In case the axis is parallel to the $y$ axis, interchange $x$ and $y$ or replace $\theta$ by $\frac{1}{2}\pi - \theta$ [or $90^\circ - \theta$].

How to create parabola

parabola - animated

Hyperbola with center $C(x_0 \textrm{ , } y_0)$ and major axis parallel to $x$ axis

Hyperbola
Length of major axis $A'A = 2a$
Length of minor axis $B'B = 2b$
Distance from center $C$ to focus $F$ or $F'$ is
$c = \sqrt{a^2 + b^2}$
Eccentricity = $\epsilon = \frac{c}{a} = \frac{\sqrt{a^2 + b^2}}{a}$
Equation in rectangular coordinates:
$\frac{(x - x_0)^2}{a^2} - \frac{(y - y_0)^2}{b^2} = 1$
Slopes of asymptotes $G'H$ and $GH' = \pm \frac{b}{a}$
Equation in polar coordinates if $C$ is at $O$:
$r^2 = \frac{a^2b^2}{b^2 \textrm{ cos }^2 \theta - a^2 \textrm{ sin }^2 \theta}$
Equation in polar coordinates if $C$ is on $X$ axis and $F'$ is at $O$:
$r = \frac{a(c^2 - 1)}{1 - \epsilon \textrm{ cos } \theta}$
If $P$ is any point on the hyperbola, $PF - PF' = \pm 2a$ [depending on branch]
If the major axis is parallel to the $y$ axis, interchange $x$ and $y$ in the above or replace $\theta$ by $\frac{1}{2}\pi - \theta$ [or $90^\circ - \theta$].

How to create hyperbola

hyperbola - animation


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