To each conic section (ellipse, parabola, hyperbola, circle) there is a number called the eccentricity that uniquely characterizes the shape of the curve. Conic Sections: Eccentricit
Mar 27, 2015
To each conic section (ellipse, parabola, hyperbola, circle) there is a number called the eccentricity that uniquely characterizes the shape of the curve.
Conic Sections: Eccentricity
If e = 1, the conic is a parabola.
If e = 0, the conic is a circle.
If e < 1, the conic is an ellipse.
If e > 1, the conic is a hyperbola.
Conic Sections: Eccentricity
For both an ellipse and a hyperbola
eca
where c is the distance from the center to the focus and a is the distance from the center to a vertex.
Conic Sections: Eccentricity
Classifying Conics10.6
What is the general 2nd degree equation for any conic?
What information can the discriminant tell you about a conic?
The equation of any conic can be written
in the form-
Called a general 2nd degree equation
2 2 0Ax Bxy Cy Dx Ey F
Circles
Can be multiplied out to look like this….
2 2( 1) ( 2) 16x y
2 2 2 4 11 0x y x y
Ellipse
Can be written like this…..
22( 1)
( 1) 14
xy
2 24 2 8 1 0x y x y
Parabola
Can be written like this…..
2( 6) 4( 8)y x
2 12 4 4 0y y x
Hyperbola
Can be written like this…..
22 ( 4)
( 4) 19
yx
2 29 72 8 1 0x y x y
How do you know which conic it is when it’s been multiplied
out?
• Pay close attention to whose squared and whose not…
• Look at the coefficients in front of the squared terms and their signs.
Circle Both x and y are
squared
And their coefficients are the same number and sign
2 2 2 4 11 0x y x y
Ellipse• Both x and y are
squared• Their coefficients are
different but their signs remain the same.
2 24 2 8 1 0x y x y
Parabola• Either x or y is
squared but not both
2 12 4 4 0y y x
Hyperbola
• Both x and y are squared
• Their coefficients are different and so are their signs.
2 29 72 8 1 0x y x y
You Try!
0343.10
036164.9
0782.8
0593033.7
0164y8x2x6.
046y6x2y2x.5
0314y12x2y2x4.
03023y25x3.
041y20x22x2.
032x24y2x1.
22
22
22
22
yyx
yxyx
xyx
xyx
1.Ellipse
2.Parabola
3.Hyperbola
4.Circle
5.Hyperbola
6.Parabola
7.Circle
8.Ellipse
9.Hyperbola
10.Ellipse
When you want to be sure…
of a conic equation, then find the type of conic using discriminate information:
Ax2 +Bxy +Cy2 +Dx +Ey +F = 0
B2 − 4AC < 0, B = 0 & A = C Circle
B2 − 4AC < 0 & either B≠0 or A≠C Ellipse
B2 − 4AC = 0 Parabola
B2 − 4AC > 0 Hyperbola
Classify the Conic
2x2 + y2 −4x − 4 = 0
Ax2 +Bxy +Cy2 +Dx +Ey +F = 0
A = 2
B = 0
C = 1
B2 − 4AC = 02 − 4(2)(1) = −8
B2 − 4AC < 0, the conic is an ellipse
Graph the Conic2x2 + y2 −4x − 4 = 0
2x2 −4x + y2 = 4
2(x2 −2x +___)+ y2 = 4 + ___ (−2/2)2= 1
2(x2 −2x +1)+ y2 = 4 + 2(1)
2(x−1)2 + y2 = 6
V(1±√6), CV(1±√3)
166
)1(2 22
yx
163
)1( 22
yx
Complete the Square
Steps to Complete the Square1. Group x’s and y’s. (Boys with the boys and
girls with the girls) Send constant numbers to the other side of the equal sign.
2. The coefficient of the x2 and y2 must be 1. If not, factor out.
3. Take the number before the x, divide by 2 and square. Do the same with the number before y.
4. Add these numbers to both sides of the equation. *(Multiply it by the common factor in #2)
5. Factor
Write the equation in standard form by completing the square
01824 22 yxyx
______1___)2(4___2 22 yyxx
1)2(42 22 yyxx
)1)(4(11)12(4)12( 22 yyxx
4)1(4)1(22 yx
4
4
4
)1(4
4
)1( 22
yx
11
)1(
4
)1( 22
yx
12
22
What is the general 2nd degree equation for any conic?
What information can the discriminant tell you about a conic?
B2- 4AC < 0, B = 0, A = CCircle
B2- 4AC < 0, B ≠ 0, A ≠ CEllipse
B2- 4AC = 0, Parabola
B2- 4AC > 0 Hyperbola
2 2 0Ax Bxy Cy Dx Ey F
Assignment 10.6
Page 628, 29-55 odd