simple explanation of focus of parabola, foci of hyperbola?

[georgebaseball]

Can somebody please explain me -- with easy words -- what are the focus of a parabola and the foci of a hyperbola? My book defines a foci as a fixed point, but I don't understand. For example, how can you locate visually the focus of a parabola? I can locate visually the vertex, x intercepts, y intercepts, but I don't know how to locate the focus.

Thank you!

 

[tkhunny]

That's a bit too broad. Just a few notes.

A definition of a parabola refers to a locus of points equidistant from a point and a line. That point is the focus.

A definition of an ellipse refers to a locus of points with constant sum of distance from two points. These points are the foci.

A definition of an hyperbola refers to a locus of points with constant difference of distances from two points. These points are the foci.

 

[stapel]

Draw a straight line. Draw a dot off to the side.

Draw a dot exactly midway between the dot and the line.

Go off to the side a bit. Draw a line segment perpendicular to the original line, heading up past the midway dot. Whatever length you drew the segment, draw a circle with the same radius, around that first dot. The segment should just meet the circle. Draw another dot there.

This last circle-and-segment dot is, necessarily, equidistant from the original straight line and the original dot. If you continue on in this manner (it's a slow process), you will eventually end up with a parabola.

The original straight line is the directrix. The first dot is the focus. The second dot is the vertex.

Eliz.