Conic Sections Case Study

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Degenerate Cases of Conic Sections: Conics are formed when a plane and a double cone intersect. These intersections can form a variety of conics. These include: circles, parabolas, ellipses and hyperbolas. A circle can be formed from a cone if a plane intersects with the cone at an angle perpendicular to the base of the cone. This circle is also formed by the equation: (x-h)2+ (y-k)2=r2. This equation creates a relation because the collection of points does not pass the vertical line test. To form a specific circle, only 3, non-collinear, unique points are necessary. In order to form a circle when B is equal to 0, then A must equal C. If this intersecting plane cuts the cone at any angle less than or greater than 90o, yet still excluding 90o it will form an ellipse. An ellipse is a locus of points …show more content…
It's equation is: (x-h)2a2+(y-k)2b2=1. AC> 0to create an ellipse when B = 0. Additionally, if the plane cuts through the cone at an angle that is the same as the surface of the cone, it will from a parabola. A parabola can be considered as a locus of points that are each equidistant from a directrix, which is a given line, and a focus, which is a given point. If B = 0, then A must equal 0 or C must equal 0 but not both. The equation of a horizontal hyperbola is: y-k=14p(x-h)2and x-h=14p(y-k)2for a vertical ellipse. In this situation, the a value is 14p. Finally, a hyperbola can be formed by tilting the plane at an angle much higher than the surface of the cone. This creates the equation (x-h)2a2-(y-k)2b2=1for horizontal hyperbolas and (y-k)2a2 -(x-h)2b2=1for vertical hyperbolas. A hyperbola is a locus of points with a fixed difference from the two given foci. This distance between the hyperbola and the foci is given. AC

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