Make a unit circle and a unit hyperbola That is the graph of

Make a unit circle and a unit hyperbola. That is, the graph of

and the graph of

The unit hyperbola has the diagonal lines with equations

and

as asymptotes.

Solution

A parabola is \"the set of all points in a plane equidistant from a fixed point (focus) and a fixed line (directrix)\".

The distances to any point (x,y) on the parabola from the focus (0,p) and the directrix y=-p, are equal to each other. This can be used to develop the equation of a parabola.

If you take the definition of a parabola and work out the algebra, you can develop the equation of a parabola. You can click on the link if you\'d like to see the development, but the short version is that the standard form is x² = 4py.

The parabola has the property that any signal (light, sound, etc) entering the parabola parallel to the axis of symmetry will be reflected through the focus (this is why satellite dishes and those parabolic antennas that the detectives use to eavesdrop on conversations work). Also, any signal originating at the focus will be reflected out parallel to the axis of symmetry (this is why flashlights work).