Conic Sections
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Conics are plane curves formed by cutting a double right circular cone with a plane. The type of conic section depends on the angle of the cutting plane to the cone's axis: a circle for perpendicular, ellipse for non-perpendicular, parabola for parallel to the edge, and hyperbola for intersecting both cones. A parabola is the set of points equidistant from a focus point and directrix line, with the vertex halfway between them. The latus rectum connects the endpoints equidistant from the focus, determining how wide the parabola opens. The four types of parabolas depend on the axis and opening direction.
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Conic Sections
- 1. Conics
- 2. DEFINITION Conic sections are plane curves that can be formed by cutting a double right circular cone with a plane at various angles.
- 3. AXIS DOUBLE RIGHT CIRCULAR CONE A circle is formed when the plane intersects one cone and is perpendicular to the axis
- 4. An ellipse is formed when the plane intersects one cone and is NOT perpendicular to the axis.
- 5. A parabola is formed when the plane intersects one cone and is parallel to the edge of the cone.
- 6. A hyperbola is formed when the plane intersects both cones.
- 8. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. It can be defined as the locus of points whose distances are in a fixed ratio to some point, called a focus, and some line, called a directrix.
- 9. GENERAL EQUATION OF CONICS 𝑨𝒙𝟐+𝑩𝒙𝒚+𝑪𝒚𝟐+𝑫𝒙+𝑬𝒚+𝑭=𝟎 DISCRIMINANT Ellipse Parabola Hyperbola 𝑩𝟐−𝟒𝑨𝑪<𝟎 𝑩𝟐−𝟒𝑨𝑪=𝟎 𝑩𝟐−𝟒𝑨𝑪>𝟎
- 10. Parabola: A = 0 or C = 0 Circle: A = C Ellipse: A = B, but both have the same sign Hyperbola: A and C have Different signs
- 11. The Parabola The parabolais a set of points which are equidistant from a fixed point (the focus) and the fixed line (the directrix).
- 12. PROPERTIES The line through the focus perpendicular to the directrix is called the axis of symmetry or simply the axis of the curve. 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.
- 14. The line through F perpendicular to the axis is called the latus rectum whose length is |4a|. The endpoints are 𝑳𝟏and𝑳𝟐. This determines how the wide the parabola opens.
- 15. The line parallel to the latus rectum is called the directrix.
- 16. 𝑳𝟏 𝑷(𝒙,𝒚) Directrix Latus Rectum abr />|a| Vertex Focus Axis of Symmetry 𝑳𝟐
- 18. 𝑽(𝟎,𝟎) 𝑳𝟏(𝒂,𝟐𝒂) 𝒂𝒙𝒊𝒔: 𝒙 𝒐𝒑𝒆𝒏𝒊𝒏𝒈: 𝒕𝒐 𝒕𝒉𝒆 𝒓𝒊𝒈𝒉𝒕 𝑭(𝒂,𝟎) 𝑳𝟐(𝒂,−𝟐𝒂) 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏: 𝒚𝟐=𝟒𝒂𝒙 𝑫:𝒙=−𝒂 TYPE 1
- 19. 𝑽(𝟎,𝟎) 𝑳𝟏(−𝒂,𝟐𝒂) 𝒂𝒙𝒊𝒔: 𝒙 𝒐𝒑𝒆𝒏𝒊𝒏𝒈: 𝒕𝒐 𝒕𝒉𝒆 𝒍𝒆𝒇𝒕 𝑭(−𝒂,𝟎) 𝑳𝟐(−𝒂,−𝟐𝒂) 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏: 𝒚𝟐=−𝟒𝒂𝒙 𝑫:𝒙=𝒂 TYPE 2
- 20. 𝑽(𝟎,𝟎) 𝑳𝟏(𝟐𝒂,𝒂) 𝒂𝒙𝒊𝒔: 𝒚 𝒐𝒑𝒆𝒏𝒊𝒏𝒈: 𝒖𝒑𝒘𝒂𝒓𝒅 𝑭(𝟎,𝒂) 𝑳𝟐(−𝟐𝒂,𝒂) 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏: 𝒙𝟐=𝟒𝒂𝒚 𝑫:𝒚=−𝒂 TYPE 3
- 21. 𝑽(𝟎,𝟎) 𝑳𝟏(−𝟐𝒂,−𝒂) 𝒂𝒙𝒊𝒔: 𝒚 𝒐𝒑𝒆𝒏𝒊𝒏𝒈:𝒅𝒐𝒘𝒏𝒘𝒂𝒓𝒅 𝑭(𝟎,−𝒂) 𝑳𝟐(𝟐𝒂,−𝒂) 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏: 𝒙𝟐=−𝟒𝒂𝒚 𝑫:𝒚=𝒂 TYPE 4
- 22. Sample Problem Locate the coordinates of the vertex (V), focus (F), endpoints of the latus rectum (𝑳𝟏𝑳𝟐), the equation of the directrix, and sketch the graph of 𝒙𝟐=−𝟖𝒚.
- 23. solution 1. 𝒙𝟐=−𝟖𝒚 takes the form 𝒙𝟐=−𝟒𝒂𝒚 2. the parabola opens downward 3. Compute the value of 𝒂 4. so, −𝟒𝒂=−𝟖, or 𝒂=𝟐 5. the required coordinates are 𝑽(𝟎,𝟎) 𝑫:𝒚=𝒂 𝑭𝟎,−𝒂=𝑭(𝟎,−𝟐) 𝑫:𝒚=𝟐 𝑳𝟏−𝟐𝒂,−𝒂=𝑳𝟏(−𝟒,−𝟐) 𝑳𝟐𝟐𝒂,−𝒂=𝑳𝟐(𝟒, −𝟐)
- 24. 𝒚 | | | 1 2 3 𝒚=𝟐 𝑽(𝟎,𝟎) 𝒙 | | | | | -5 -4 -3 -2 -1 | | | | | 1 2 3 4 5 | | | -3 -2 -1 𝑳𝟏(−𝟒,−𝟐) 𝑳𝟐(𝟒,−𝟐) 𝑭(𝟎,−𝟐)
- 25. Sketch the graphs and determine the coordinates of V, F, ends of LR, and equation of the directrix. 1. 𝒙𝟐+𝟔𝐲=𝟎 2. 𝒚𝟐=−𝟐𝟒𝒙 3. 𝟐𝒚𝟐−𝟑𝒙=𝟎