A circle is the set of all points equidistant from a fixed point (center). The standard form equation is:
\[(x - h)^2 + (y - k)^2 = r^2\]
where \((h,k)\) is the center and \(r\) is the radius.
An ellipse is the set of points where the sum of distances from two fixed points (foci) is constant. The standard form equation is:
\[\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\]
where \((h,k)\) is the center, \(a\) is the length of the semi-major axis, and \(b\) is the length of the semi-minor axis.
A hyperbola is the set of points where the difference of distances from two fixed points (foci) is constant. The standard form equations are:
For horizontal transverse axis: \[\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\]
For vertical transverse axis: \[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1\]
A parabola is the set of points equidistant from a fixed point (focus) and a fixed line (directrix). The standard form equations are:
Vertical axis: \[(x - h)^2 = 4p(y - k)\]
Horizontal axis: \[(y - k)^2 = 4p(x - h)\]
where \((h,k)\) is the vertex and \(|p|\) is the distance from vertex to focus.
Test your understanding of all conic sections.