Sections of a cone, circle, ellipse, parabola and hyperbola and Degenerate Conic Sections Video Lecture | Mathematics (Maths) for JEE Main & Advanced

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
vortex 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 1 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 map
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
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
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
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 coal hyperbola the plane
intersects only one map 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 vertical axis at an angle smaller
than the vertex angle the plane
intersects both the Knapp's of the cone
to form an open curve called a hyperbola
which has two disjoint curves degenerate
comix if the plane intersects the double
right circular cone at its vertex the
ellipse becomes a point the parabola
becomes aligned and the hyperbola
becomes two intersecting lines the
figure so obtained are called degenerate
connects in this module you have learned
that when a plane intersects a double
white 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
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