The Conics
By: Diana Brown
Day Ten:
Summary
Conic
Sections
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|
|
|
Circle |
Ellipse (h) |
Parabola (h) |
Hyperbola (h) |
Definition: |
Ellipse (v) |
Parabola (v) |
Hyperbola (v) |
By changing the angle and location of intersection, we can produce a circle,
ellipse, parabola or hyperbola; or in the special case when the plane touches
the vertex: a point, line or 2 intersecting lines.
|
|
|
|
|
Point |
Line |
Double Line |
The General Equation for a Conic Section: |
The type of section can be found from the sign of: B2
- 4AC
If B2 - 4AC is... |
then the curve is a... |
< 0 |
ellipse, circle, point or no curve. |
= 0 |
parabola, 2 parallel lines, 1 line or no curve. |
> 0 |
hyperbola or 2 intersecting lines. |
The Conic Sections. For any of the below with a
center (j, k) instead of (0, 0), replace each x term with (x-j) and each
y term with (y-k).
|
Circle |
Ellipse |
Parabola |
Hyperbola |
Equation (horiz. vertex): |
x2 + y2 = r2 |
x2 / a2 + y2 / b2 =
1 |
4px = y2 |
x2 / a2 - y2 / b2 =
1 |
Equations of Asymptotes: |
|
|
|
y = ± (b/a)x |
Equation (vert. vertex): |
x2 + y2 = r2 |
y2 / a2 + x2 / b2 =
1 |
4py = x2 |
y2 / a2 - x2 / b2 =
1 |
Equations of Asymptotes: |
|
|
|
x = ± (b/a)y |
Variables: |
r = circle radius |
a = major radius (= 1/2 length major axis) |
p = distance from vertex to focus (or directrix) |
a = 1/2 length major axis |
Eccentricity: |
0 |
|
c/a |
c/a |
Relation to Focus: |
p = 0 |
a2 - b2 = c2 |
p = p |
a2 + b2 = c2 |
Definition: is the locus of all points which meet the
condition... |
distance to the origin is constant |
sum of distances to each focus is constant |
distance to focus = distance to directrix |
difference between distances to each foci is constant |
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