(10.5) Conic Sections
by Eleanor H.
What are the conic shapes?
Circle (not shown)
Ellipse (drawn above)
Parabola (drawn above)
Hyperbola (drawn above)
Point
Line
Intersecting Lines
What is eccentricity? It is the ratio of the distance between any point of a conic section and a fixed point to the distance between the same point of the conic section to a fixed line.
The General Equation for Conic Sections is Ax2 + Bx + Cy2 + Ey + F = 0 (A, B, C are not all 0)
| Conic Section | Standard Form of Equation | Variations of forms |
| circle | (x-h)2 + (y-k)2 = r2 | A = C |
| parabola | (y-k)2 = 4p(x-h) or
(x-h) 2= 4p(y-k) |
Either A or C is zero. |
| ellipse | (x-h)2/a2
+ (y-k)2/b2
= 1 or
(y-k)2/a2 + (x-h)2/b2 = 1 |
A and C have the same sign; A and C are not equal. |
| hyperbola | (x-h)2/a2
- (y-k)2/b2
= 1 or
(y-k)2/a2 - (x-h)2/b2 = 1 or xy = k |
A and C have opposite signs. |
Word Problem for section 10.5:
Madison and Jane were dying Easter eggs with many different colors. This time, instead of boiling the egg first, they decide to create a hole in both ends of major axis in order to blow out the yolk. These two points are on the major axis. The center of the egg is at (0,0) and the two major axis points are at (-3,0) and (3,0).
a) name the eccentricity (e) of this ellipse when the equation is (x)2/9 + (y)2/4 = 1
Answers and Steps:
a) step 1: the equation for the eccentricity of and ellipse is c/a = e
c = the distance from the center to the focus
a = the distance from the center to the vertex (found in standard form equations from above)
step 2: in order to get (c), the foci must be found first.
Ö(big-small) The square root off big - small is found from the denominators. This gives the distance from the center to the foci.
Ö(9-4) = Ö5
The foci are (-Ö5,0) and (Ö5,0). (found by using ± 5 from x of center coordinates)
step 3: Find the distance from the center to one of the foci, using the distance formula.
Ö(0-0)2 + (0- Ö5)2 = Ö5 = c
step 4: Plug in c and a into the equation of eccentricity.
a is found from using the formula ( Ö9 = 3)
Ö5/3 = .7454 = e
The eccentricity of the ellipse (or Easter egg) is .7454.
QUOTE:
"Never be bullied into silene.
Never alow yourself to be made a victim. Accept no one's definition of
your life: define yourself." by Robert Fierstein
--This quote describes the character traits of integrity, courage, kindness, good judgement, self-discipline, perseverance. All of these are great qualities that help describe one's own self, becoming a unique individual.
The images and graphics are from the websites www.school.discovery.com and www.bellsnwhistles.com.