can one of the curves of a hyperbola be a parabola?

Question

give me some reference sites

Answer 1

The answer is NO. A parabola and a hyperbola are both conic sections which means they are formed when a plane intersects a cone. However, the conditions in which they are formed are different.

If the plane intersects the axis at an angle equal to the angle the cone makes with the axis, the intersection will form a parabola. If the plane intersects the axis at an angle smaller than the angle the cone makes with the axis, the intersection will form a hyperbola. They equations are different and the curves will never equal each other.

Even if you consider a degenerate hyperbola and parabola the curve of a parabola will NOT equal one of the curves of a hyperbola.

A line (degenerate parabola) is formed when the plane passes through the vertex and along the sides of the two cones. On the other hand, Intersecting lines (degenerate hyperbola) will form when a plane passes through the vertex of the the two cones at an angle less than the angle of the cones. In that case one of the curves of a degenerate hyperbola would look like two lines meeting at an angle while a degenerate parabola would simply look like a line.

Answer 2

Yes. Using conic sections, if the cone is cut by the plane through the intersection of the upper and lower cones, such that the plane is tangential to both upper and lower cone, the intersection will be a straight line. That line can be described by both the equations of a parabola or the equations of a hyperbola. For a parabola y=ax^2 then a is infinity. The equation for the hyperbola is treated similarly.

Answer 3

no. by definition, (the way the 2 cones are cut to create either a hyperbola or parabola) the two halves of a hyperbola are different from any parabola

Answer 4

no. different definitions, different equations, different properties.