#
Conic Sections Solver

Input the equation of a parabola, circle, ellipse, or hyperbola to see
pertinent information about the conic, including vertices, asymptotes,
foci, and more. The Pro version unlocks solutions to conics shifted away from the origin.

Parabolas Circles and Ellipses
Hyperbolas

### Parabolas

Standard Form:
\[x^2 = 4py\]
\[y^2 = 4px\]
A **parabola** is a curve on which every point is equidistant from a
point - the **focus** - and a line - the **directrix**. Only one
variable is squared in the equation of a parabola. If y is squared, the
parabola opens either left or right, depending on the sign of p. If x is
squared, the parabola opens either up or down. The value of p is the distance
from the **vertex** of the parabola to the focus. The directrix is the
same distance away from the vertex, in the opposite direction.

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### Circles and Ellipses

Standard Form:
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
\[\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\]
The equation of a **circle** or **ellipse** has both x and y squared
and one is added to the other. The value of $a^2$
is larger than the value of $b^2$.
If $a^2$ is underneath the
$x^2$ term, the ellipse will have
a horizontal **major axis**. Otherwise, the ellipse has a vertical major
axis. The value of a is the distance from the **center** of the ellipse
to either **vertex**. The value of b is the distance from the center to
either **covertex**. A circle is a special type of ellipse in which a
and b are equal. This value is the **radius**, r. The sum of the
distances from any point on the ellipse to the **foci** is constant.
The foci are found by solving the equation
\[c^2 = a^2 - b^2\]
The value of c is the distance from the center of the ellipse
to either focus. The foci are found on the major axis.

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### Hyperbolas

Standard Form:
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
\[\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \]
The equation of a **hyperbola** has both x and y squared and one is
subtracted from the other. If the $y^2$ term comes first, the hyperbola
has a vertical **transverse axis**, meaning it opens up and down, and
the value of b is the distance from the center to either **vertex**.
If the $x^2$ term comes first, the hyperbola has a horizontal transverse
axis - it opens left and right and the value of a is the distance from the
center to either vertex. The differences between the distances from any
point on a hyperbola to the two **foci** are constant. The foci are
found by solving the equation
\[c^2 = a^2 + b^2\]
The value of c is the distance from the center of the hyperbola to either
focus. The foci are found on the transverse axis. The **asymptotes**
indicate the long run behavior of the hyperbola. The slopes of the
asymptotes are found by calculating
$\pm\frac{b}{a}$

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