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# Conic Sections

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### Conic Sections

1. 1. Conics<br />
2. 2. DEFINITION<br />Conic sections are plane curves that can be formed by cutting a double right circular cone with a plane at various angles. <br />
3. 3. AXIS<br />DOUBLE RIGHT CIRCULAR CONE<br />A circle is formed when the plane intersects one cone and is perpendicular to the axis<br />
4. 4. An ellipse is formed when the plane intersects one cone and is NOT perpendicular to the axis.<br />
5. 5. A parabola is formed when the plane intersects one cone and is parallel to the edge of the cone.<br />
6. 6. A hyperbola is formed when the plane intersects both cones.<br />
7. 7. DEGENERATE CONIC<br />
8. 8. InΒ analytic geometry, a conic may be defined as aΒ plane algebraic curveΒ of degree 2. <br /> It can be defined as theΒ locusΒ of points whose distances are in a fixed ratio to some point, called aΒ focus, and some line, called aΒ directrix.<br />
9. 9. GENERAL EQUATION OF CONICS<br />π¨ππ+π©ππ+πͺππ+π«π+π¬π+π­=π<br />Β <br />DISCRIMINANT<br />Ellipse<br />Parabola<br />Hyperbola<br />π©πβππ¨πͺ<π<br />Β <br />π©πβππ¨πͺ=π<br />Β <br />π©πβππ¨πͺ>π<br />Β <br />
10. 10. Parabola: A = 0 or C = 0<br />Circle: A = C<br />Ellipse: A = B, but both have the same sign <br />Hyperbola: A and C have Different signs<br />
11. 11. The Parabola<br />The parabolais a set of points which are equidistant from a fixed point (the focus) and the fixed line (the directrix).<br />
12. 12. PROPERTIES<br />The line through the focus perpendicular to the directrix is called the axis of symmetry or simply the axis of the curve.<br />The point where the axis intersects the curve is the vertex of the parabola. The vertex (denoted by V) is a point midway between the focus and directrix.<br />
13. 13. <ul><li>The undirected distance from V to F is a positive number denoted by |a|.
14. 14. The line through F perpendicular to the axis is called the latus rectum whose length is |4a|. The endpoints are π³πandπ³π. This determines how the wide the parabola opens.
15. 15. The line parallel to the latus rectum is called the directrix.</li></ul>Β <br />
16. 16. π³π<br />Β <br />π·(π,π)<br />Β <br />Directrix<br />Latus Rectum 4a<br />|a|<br />Vertex<br />Focus<br />Axis of Symmetry<br />π³π<br />Β <br />
17. 17. TYPES OF PARABOLA<br />
18. 18. π½(π,π)<br />Β <br />π³π(π,ππ)<br />Β <br />ππππ:Β π<br />Β <br />πππππππ:Β ππΒ πππΒ πππππ<br />Β <br />π­(π,π)<br />Β <br />π³π(π,βππ)<br />Β <br />ππππππππ:Β ππ=πππ<br />Β <br />π«:π=βπ<br />Β <br />TYPE 1<br />
19. 19. π½(π,π)<br />Β <br />π³π(βπ,ππ)<br />Β <br />ππππ:Β π<br />Β <br />πππππππ:Β ππΒ πππΒ ππππ<br />Β <br />π­(βπ,π)<br />Β <br />π³π(βπ,βππ)<br />Β <br />ππππππππ:Β ππ=βπππ<br />Β <br />π«:π=π<br />Β <br />TYPE 2<br />
20. 20. π½(π,π)<br />Β <br />π³π(ππ,π)<br />Β <br />ππππ:Β π<br />Β <br />πππππππ:Β Β ππππππ<br />Β <br />π­(π,π)<br />Β <br />π³π(βππ,π)<br />Β <br />ππππππππ:Β ππ=πππ<br />Β <br />π«:π=βπ<br />Β <br />TYPE 3<br />
21. 21. π½(π,π)<br />Β <br />π³π(βππ,βπ)<br />Β <br />ππππ:Β π<br />Β <br />πππππππ:ππππππππ<br />Β <br />π­(π,βπ)<br />Β <br />π³π(ππ,βπ)<br />Β <br />ππππππππ:Β ππ=βπππ<br />Β <br />π«:π=π<br />Β <br />TYPE 4<br />
22. 22. Sample Problem<br />Locate the coordinates of the vertex (V), focus (F), endpoints of the latus rectum (π³ππ³π), the equation of the directrix, and sketch the graph of ππ=βππ.<br />Β <br />
23. 23. solution<br />1. ππ=βππ takes the form ππ=βπππ<br />2. the parabola opens downward<br />3. Compute the value of π<br />4. so, βππ=βπ, or π=π<br />5. the required coordinates are<br />Β <br />π½(π,π)<br />Β <br />π«:π=π<br />Β <br />π­π,βπ=π­(π,βπ)<br />Β <br />π«:π=π<br />Β <br />π³πβππ,βπ=π³π(βπ,βπ)<br />Β <br />π³πππ,βπ=π³π(π,Β βπ)<br />Β <br />
24. 24. π<br />Β <br /> | | | <br /> 1 2 3 <br />π=π<br />Β <br />π½(π,π)<br />Β <br />π<br />Β <br /> | | | | |<br /> -5 -4 -3 -2 -1<br /> | | | | |<br /> 1 2 3 4 5<br /> | | | <br />-3 -2 -1 <br />π³π(βπ,βπ)<br />Β <br />π³π(π,βπ)<br />Β <br />π­(π,βπ)<br />Β <br />
25. 25. Sketch the graphs and determine the coordinates of V, F, ends of LR, and equation of the directrix.<br />1. ππ+ππ²=π<br />2. ππ=βπππ<br />3. πππβππ=π<br />Β <br />