This is a free lesson from our course in Algebra II
A conic section is the
intersection of a
plane and a
cone. By changing the
angle and location of intersection, you
can produce a
circle,
ellipse,
parabola or
hyperbola. In general the conic section
is expressed by second
degree
polynomial in two variables x and y:
as Ax^{2} + Bxy + Cy^{2} + Dx
+ Ey + F = 0 and the 'Type' label displays what type of conic section you
see in the graph, can be determined by the value of the discriminant
B^{2} 4AC. Remember it:
If B^{2}  4AC is:
• < 0, the curve is ellipse, circle, point or no curve.
• = 0, it is parabola, two parallel lines, 1 line or no curve.
• > 0, it is hyperbola or two intersecting lines.
(More text below video...)
(Continued from above) Apart from above geometric notation to each of these curves, algebraically it can
be expressed using key notion 'eccentricity'. If conic section explained by relationship
of a line in the plane called the directrix and focus, it is 'set of all points whose distance
to the focus is a constant times the distance to the directrix'. This constant is
called eccentricity and expressed as a number that uniquely characterizes the shape
of the curve. It is denoted by the letter e.
Remember for:
ellipse
e <1
circle e
= 0
parabola
e = 1
hyperbola
e > 1
You'll see that there is only one general kind of curve called a conic, and others
are special cases depending on the conic's eccentricity. In the above polynomial
having variables x and y, 'xy' term can be eliminated
by a 'rotation of axes'. Then complete the square with respect to both x
and y, and you'll get one of the standard equations for the sections
like hyperbola, parabola etc. To graph, first identify whether the given equation
is a circle, a parabola or a hyperbola and then find all the special points such
as center, vertex, foci and asymptotes and then graph them.
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