perpendicular distance is the exact same distance from a parabola.
what is the exact same distance from a parabola?
perpendicular distance
in the diagram above, the point v is the foot of the perpendicular from the vertex of the parabola to the axis of the cone. the point f is the foot of the perpendicular from the point v to the plane of the parabola. by symmetry, f is on the axis of symmetry of the parabola. angle vpf is complementary to , and angle pvf is complementary to angle vpf, therefore angle pvf is . since the length of is , the distance of f from the vertex of the parabola is . it is shown above that this distance equals the focal length of the parabola, which is the distance from the vertex to the focus. the focus and the point f are therefore equally distant from the vertex, along the same line, which implies that they are the same point. therefore, the point f, defined above, is the focus of the parabola.
parabola
a parabola formula_66 with vertex formula_24 can be transformed by the translation formula_68 to one with the origin as vertex. a suitable rotation around the origin can then transform the parabola to one that has the -axis as axis of symmetry. hence the parabola formula_66 can be transformed by a rigid motion to a parabola with an equation formula_70. such a parabola can then be transformed by the uniform scaling formula_71 into the unit parabola with equation formula_72. thus, any parabola can be mapped to the unit parabola by a similarity.
parabola