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Ÿ+π‘©π’™π’š+π‘ͺπ’šπŸ+𝑫𝒙+π‘¬π’š+𝑭=𝟎<br />Β <br />DISCRIMINANT<br />Ellipse<br />Parabola<br />Hyperbola<br />π‘©πŸβˆ’πŸ’π‘¨π‘ͺ<𝟎<br />Β <br />π‘©πŸβˆ’πŸ’π‘¨π‘ͺ=𝟎<br />Β <br />π‘©πŸβˆ’πŸ’π‘¨π‘ͺ>𝟎<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Ÿandπ‘³πŸ. 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Ÿ<br />Β <br />𝑷(𝒙,π’š)<br />Β <br />Directrix<br />Latus Rectum 4a<br />|a|<br />Vertex<br />Focus<br />Axis of Symmetry<br />π‘³πŸ<br />Β <br />
  17. 17. TYPES OF PARABOLA<br />
  18. 18. 𝑽(𝟎,𝟎)<br />Β <br />π‘³πŸ(𝒂,πŸπ’‚)<br />Β <br />π’‚π’™π’Šπ’”: 𝒙<br />Β <br />π’π’‘π’†π’π’Šπ’π’ˆ:Β π’•π’Β π’•π’‰π’†Β π’“π’Šπ’ˆπ’‰π’•<br />Β <br />𝑭(𝒂,𝟎)<br />Β <br />π‘³πŸ(𝒂,βˆ’πŸπ’‚)<br />Β <br />π’†π’’π’–π’‚π’•π’Šπ’π’:Β π’šπŸ=πŸ’π’‚π’™<br />Β <br />𝑫:𝒙=βˆ’π’‚<br />Β <br />TYPE 1<br />
  19. 19. 𝑽(𝟎,𝟎)<br />Β <br />π‘³πŸ(βˆ’π’‚,πŸπ’‚)<br />Β <br />π’‚π’™π’Šπ’”: 𝒙<br />Β <br />π’π’‘π’†π’π’Šπ’π’ˆ: 𝒕𝒐 𝒕𝒉𝒆 𝒍𝒆𝒇𝒕<br />Β <br />𝑭(βˆ’π’‚,𝟎)<br />Β <br />π‘³πŸ(βˆ’π’‚,βˆ’πŸπ’‚)<br />Β <br />π’†π’’π’–π’‚π’•π’Šπ’π’:Β π’šπŸ=βˆ’πŸ’π’‚π’™<br />Β <br />𝑫:𝒙=𝒂<br />Β <br />TYPE 2<br />
  20. 20. 𝑽(𝟎,𝟎)<br />Β <br />π‘³πŸ(πŸπ’‚,𝒂)<br />Β <br />π’‚π’™π’Šπ’”:Β π’š<br />Β <br />π’π’‘π’†π’π’Šπ’π’ˆ:Β Β π’–π’‘π’˜π’‚π’“π’…<br />Β <br />𝑭(𝟎,𝒂)<br />Β <br />π‘³πŸ(βˆ’πŸπ’‚,𝒂)<br />Β <br />π’†π’’π’–π’‚π’•π’Šπ’π’:Β π’™πŸ=πŸ’π’‚π’š<br />Β <br />𝑫:π’š=βˆ’π’‚<br />Β <br />TYPE 3<br />
  21. 21. 𝑽(𝟎,𝟎)<br />Β <br />π‘³πŸ(βˆ’πŸπ’‚,βˆ’π’‚)<br />Β <br />π’‚π’™π’Šπ’”:Β π’š<br />Β <br />π’π’‘π’†π’π’Šπ’π’ˆ:π’…π’π’˜π’π’˜π’‚π’“π’…<br />Β <br />𝑭(𝟎,βˆ’π’‚)<br />Β <br />π‘³πŸ(πŸπ’‚,βˆ’π’‚)<br />Β <br />π’†π’’π’–π’‚π’•π’Šπ’π’:Β π’™πŸ=βˆ’πŸ’π’‚π’š<br />Β <br />𝑫:π’š=𝒂<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Ÿπ‘³πŸ), the equation of the directrix, and sketch the graph of π’™πŸ=βˆ’πŸ–π’š.<br />Β <br />
  23. 23. solution<br />1. π’™πŸ=βˆ’πŸ–π’š takes the form π’™πŸ=βˆ’πŸ’π’‚π’š<br />2. the parabola opens downward<br />3. Compute the value of 𝒂<br />4. so, βˆ’πŸ’π’‚=βˆ’πŸ–, or 𝒂=𝟐<br />5. the required coordinates are<br />Β <br />𝑽(𝟎,𝟎)<br />Β <br />𝑫:π’š=𝒂<br />Β <br />π‘­πŸŽ,βˆ’π’‚=𝑭(𝟎,βˆ’πŸ)<br />Β <br />𝑫:π’š=𝟐<br />Β <br />π‘³πŸβˆ’πŸπ’‚,βˆ’π’‚=π‘³πŸ(βˆ’πŸ’,βˆ’πŸ)<br />Β <br />π‘³πŸπŸπ’‚,βˆ’π’‚=π‘³πŸ(πŸ’,Β βˆ’πŸ)<br />Β <br />
  24. 24. π’š<br />Β <br /> | | | <br /> 1 2 3 <br />π’š=𝟐<br />Β <br />𝑽(𝟎,𝟎)<br />Β <br />𝒙<br />Β <br /> | | | | |<br /> -5 -4 -3 -2 -1<br /> | | | | |<br /> 1 2 3 4 5<br /> | | | <br />-3 -2 -1 <br />π‘³πŸ(βˆ’πŸ’,βˆ’πŸ)<br />Β <br />π‘³πŸ(πŸ’,βˆ’πŸ)<br />Β <br />𝑭(𝟎,βˆ’πŸ)<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Ÿ+πŸ”π²=𝟎<br />2. π’šπŸ=βˆ’πŸπŸ’π’™<br />3. πŸπ’šπŸβˆ’πŸ‘π’™=𝟎<br />Β <br />

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