conic sections in this module you will learn about conic sections when a straight line intersects a vertical line at a fixed point and rotates about that fixed point the surface obtained is called a double right circular cone a double right circular cone consists of two cones joined at a fixed Point called the [Music] vertex the line that rotates about the vertex is called the generator and the line that remains fixed is called the axis a right circular cone has a circular base and its axis is always perpendicular line from the center of the base to the vertex the perimeter of the base is called the directrix the lateral surface of a right circular cone is called a nap a double right circular cone has two naps the nap above the vertex is called the upper nap and that below the vertex is called the lower nap also the angle between the generator and the axis is called the vertex angle conic sections if a plane intersects a double right circular cone we get two dimensional curves of different types these curves are called conic sections [Music] depending on the angle made by the plane with the vertical axis of the cone the plane can cut the cone in three different ways [Music] ellipse when the plane intersects the double right circular cone in such a way that the angle between the plane and the axis is greater than the vertex angle we get a closed curve called an ellipse when the plane is perpendicular to the axis the ellipse becomes a circle thus a circle is a special type of ellipse [Music] Parabola when the angle made by the plane to the vertical axis is exactly equal to the vertex angle we get an open curve called a parabola at the intersecting surface of the cone [Music] hyperbola the plane intersects only one nap of the double white circular cone as long as the angle between the plane and the vertical axis is greater than or equal to the vertex angle however if the plane intersects the ver vertical axis at an angle smaller than the vertex angle the plane intersects both the Naps of the cone to form an open curve called a hyperbola which has two disjoint curves degenerate conics if the plane intersects the double right circular cone at its vertex the ellipse becomes a point the parabola becomes a line and the hyperbola becomes two intersecting lines the figures so obtained are called degenerate conics in this module you have learned that when a plane intersects a double right circular cone two dimensional curves called conic sections are formed there are three types of conic sections ellipse Parabola and hyperbola when the double right circular cone is cut by the plane so that the angle between the plane and the axis is greater than the vertex angle the ellipse is obtained when the plane is perpendicular to the axis the ellipse changes into a circle if a double right circular cone is cut by a plane so that the angle between the plane and the axis is equal to the vertex angle the parabola is formed if a double right circular cone is cut by a plane so that the angle between the plane and the axis is less than the vertex angle the hyperbola is formed if a plane intersects the double right circular cone at its vertex the figures formed are called