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Conic Sections Solver

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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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