5.1 conic sections & 5.2... · 5.1 & 5.2 conic sections and the parabola filled in...
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5.1 & 5.2 Conic Sections and the Parabola FILLED IN NOTES.notebook
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5.1 Conic Sections
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An equation of the first degree always represents a line.
The general equation of the second degree has the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
where A, B and C are not all zero.
Equations of the second degree are called conic sections. Conic sections are formed by the intersection of a plane with a right circular cone.
A cone has two portions, or nappes, separated from each other by the vertex. Some conic sections are unbounded because a cone has no base or ends thus extending infinitely far in both directions.
The traditional conic sections are the parabola, ellipse and hyperbola. A circle is a special case for the ellipse where B = 0 and A = C.
The degenerate conic sections are a pair of intersecting lines, a line, a point, and no graph at all.
No matter what cone or plane, there must be some intersection, and it cannot be a pair of parallel lines.
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THE CONICS
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THE CONICS
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In this chapter all conics will be centered on the axes.
In chapter 6 we will transform the conics so they appear anywhere in the coordinate plane
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5.2 The Parabola
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A parabola is the set of all points in a plane equidistant from a fixed point (focus) and a fixed line (directrix) not containing the focus.
Things to know:1) The vertex is equidistant to the focus and directrix. That distance
is represented by c.2) The latus rectum is the line segment through the focus with
endpoints on the parabola. The latus rectum is length 4c. Given focus (m, n) the endpoints of the latus rectum are ±2c away in the direction parallel to the directrix. If the directrix is vertical then the latus rectum endpoints are (m, n±2c). If the directrix is horizontal then the latus rectum endpoints are (m±2c, n).
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Theorem5.1
Find the set of all points equidistant from (c, 0) and x = c.
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Theorem5.2
Find the set of all points equidistant from (0, c) and y = c.
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Example 1: Sketch and discuss y2 = 8x.
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Example 2: Sketch and discuss x2 = 12y.
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Example 3: Find the equation(s) of the parabola(s) with vertex at the origin and focus (4, 0)
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