Conic Sections Research Paper

Conic sections are used all over the world. Conic sections are used in things such as bridges, roller coasters, stadiums, and other objects. A conic section is the intersection of a plane with a cone. The changes in the angle of the intersection produce a circle, ellipse, parabola, and hyperbola.

All the types of conic sections can be identified using the general form equation. The general form equation is x2+Bxy+Cy2+Dx+Ey+F=0. Using the general form equation can help identify each type of conic section. If B2-4AC < 0, if there is 2 squared terms, and if the coefficients have the same sign, but different numbers then it is an ellipse. If B2-4AC > 0, if there is 2 squared terms, and one has a negative coefficient then it is a hyperbola. If there is 2 squared terms, and both coefficients are the same then it is a circle. If B2-4AC = 0, and there is only one squared term it is a parabola.

To graph from an equation you take the h and k values and plug that in for the center, and then to get the sides of the circle you take the radius value and count that many points to get to the side.

To make an equation from a graph you take the center point and plug it in for h and k, and then you count how far one of the sides is from the center and plug that into the radius. In order to identify if it is a equation for a circle the x and y have to be squared and have the same coefficients. A parabola is easy because either x or y are squared, so only one. To identify an ellipse the equation has to have x and y squared that are positive but the coefficients are different numbers. A hyperbola equation has x and y squared and a coefficiet is negative and he other is posive Hyperbola. When x and y are both squared, and exactly one of the coefficients is negative and exactly one of the coefficients is

positive.
The first type of conic section is a circle. A conic section circle is formed when a plane is parallel to the bases. The equation for a circle with a radius and a center is (x−h)2+(y−k)2=r2 or x2+y2= r2. The h and k represent the circle’s middle or center point. The r represents the radius of the circle, and the r travels from the center of the circle outwards. The symmetries for a circle are any diameter.
Next, the second type of conic section is an ellipse. An ellipse is formed by a plane not intersecting either base. Basically it is a stretched out circle. The points are the sum of the distances from the point to the foci, which is constant. The standard equation is (x−h)2/a2 + (y−k)2/b2 =1. The h and k represent the center point. There are two types of ellipses one is short and fat and the other is tall and skinny. The short and fat ellipse has a major horizontal axis of a greater(>) than b. Meaning the foci, which is the distance from the center to the focus. The equation used to find the foci is c2=a2-b2, which utilizes the center on a major axis. The vertices are (h+-a,k), and the co-vertices are (h,k+-b). Next, the tall and skinny ellipse has a major vertical axis of a less() than 0, and if it opens down p is less() than 0, and when opening to the left p is less(<) than 0. The focus which is “p” is the inside vertex on the axis of symmetry. The directrix is the outside vertex on the axis of symmetry. Hyperbolas also have horizontal and vertical asymptotes. For horizontal the equation is y=+/- b/a(x-h)+k, and for vertical asymptotes the equation is y=+/- a/b(x-h)+k.
Overall, each of the four conic sections have a specific equation. The conic sections are circles, ellipses, hyperbolas, and parabolas. Circles, parabolas, and hyperbolas deal with a plane intersecting at least one of the bases, but each intersects at a different angle. The ellipses is the only one of the four types of conic sections that does not intersect with either base. Altogether, the four types of conic sections are seen in the real world in anything from roller coasters to a small hula hoop.