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.
13.
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.
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)
