Locus of Centers
Locus of Centers of Circle Passing Through a Point and Tangent to a Line
The moving circle below traces the parabola through its center. The circle is required to pass through the origin and be tangent with the line below. The distance of the circle's center to the origin and the its distance to the line are always equal to the radius of the circle.
The equality above satisfies the definition of a parabola in which any point in a parabola must be equidistant to a fixed point and a fixed line which are the focus and the directrix respectively. Changing the value of the slider allows one to relocate the center of the circle while still following the required conditions.
The parabolic path has a focus at and a directrix at which indicates that the parabola should be opening upwards. This means that the standard form of this parabola is given by where is the location of the vertex and is the distance between the focus and the vertex.
By observation, the distance between the focus and vertex is equal to 2 which can be easily seen by letting be equal to 0. This means that taken by subtracting to the coordinate of the focus. Substituting the know values gives the equation of the parabola above.