Parameters
Standard Form
Graph
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Conic Sections Guide
A circle is the set of all points in a plane that are equidistant from a fixed point (the center).
Standard Form: \((x - h)^2 + (y - k)^2 = r^2\)
Where \((h, k)\) is the center and \(r\) is the radius.
General Form: \(x^2 + y^2 + Dx + Ey + F = 0\)
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
Standard Forms:
- Vertical: \((x - h)^2 = 4p(y - k)\)
- Horizontal: \((y - k)^2 = 4p(x - h)\)
Where \((h, k)\) is the vertex and \(p\) is the distance from vertex to focus.
An ellipse is the set of all points where the sum of distances to two fixed points (foci) is constant.
Standard Forms:
- Horizontal major axis: \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)
- Vertical major axis: \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\)
Where \((h, k)\) is the center, \(a\) is semi-major axis, \(b\) is semi-minor axis.
A hyperbola is the set of all points where the difference of distances to two fixed points (foci) is constant.
Standard Forms:
- Horizontal transverse axis: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)
- Vertical transverse axis: \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\)
Where \((h, k)\) is the center, \(a\) is distance from center to vertex.