We all know the definition of a parabola, set of all points in a plane which have the same distance from an specific point and a line.
If we expand the idea to 3 dimensions we will see a paraboloid which is the set of all points having the same distance from an specific point and a plane here is what i mean https://en.m.wikipedia.org/wiki/File:Paraboloid_of_Revolution.svg
So my question is what is the shape of all points having the same distance from an specific point and a line in space (3 dimensions)?
The answer to my question may be an equation of such a shape or simply an image, thanks in advance!
Best Answer
Let the point be $F = (0, p, 0) $ and the line $L: (0, -p, 0) + t (1, 0, 0) $
Distance squared from the point P to the given point F is
$ PF^2 = x^2 + (y - p)^2 + z^2 $
Distance squared from point P to the line is
$ PL^2 = (x^2 + (y + p)^2 + z^2) - ( (x, y+p, z) \cdot (1, 0, 0) )^2$
which simplifies to
$ PL^2 = (y + p)^2 + z^2$
Hence the desired equation is obtained by setting $ PF^2 = PL^2 $ which gives
$ x^2 + (y - p)^2 = (y + p)^2 $
which simplifies to
$ x^2 - 2 p y = 2 p y $
i.e.
$ 4 p y = x^2 $
and this the equation of a parabolic sheet (as $z$ can assume any value).