Algebra II: Conic sections and special points
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 Ax2 + Bxy + Cy2 + 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
B2- 4AC. Remember it:
If B2 - 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...)
<h2> Conic sections and special points</h2> <p> video, AlgebraII, practice questions, quizzes, subject, math help, complex numbers, add complex numbers, subtract complex numbers, operations on complex numbers, associative law, commutative law, distributive property, example.</p> <p> To add complex numbers, add the real parts together and the imaginary parts together. Similarly, to subtract two complex numbers subtract the real parts and then subtract the imaginary parts.</p>
Other useful lessons:
Completing the square
Graphing the equation of a circle
Graphing the equation of a parabola
Graphing the equation of a hyperbola
Calculating the asymptotes
(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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