Transcript Classifying Conics

Conic Sections: Eccentricity
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
c
e
a
where c is the distance from the center
to the focus and a is the distance from
the center to a vertex.
Classifying Conics
10.6
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 form2
2
Ax  Bxy  Cy  Dx  Ey  F  0
Called a general 2nd degree
equation
Circles
( x 1)  ( y  2)  16
2
2
Can be multiplied out to look like
this….
x  y  2 x  4 y 11  0
2
2
Ellipse
( x  1)
2
 ( y  1)  1
4
2
Can be written like this…..
x  4 y  2x  8 y  1  0
2
2
Parabola
( y  6)  4( x  8)
2
Can be written like this…..
y 12 y  4x  4  0
2
Hyperbola
( y  4)
( x  4) 
1
9
2
2
Can be written like this…..
9x  y  72x  8 y  1  0
2
2
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
x  y  2 x  4 y  11  0
2
2
Ellipse
• Both x and y are
squared
• Their coefficients are
different but their
signs remain the
same.
x  4 y  2x  8 y  1  0
2
2
Parabola
• Either x or y is
squared but not both
y 12 y  4x  4  0
2
Hyperbola
• Both x and y are
squared
• Their coefficients are
different and so are
their signs.
9x  y  72x  8 y  1  0
2
2
1. x 2  4y2  2x  3  0
2. 2x2  20x  y  41  0
You Try!
1. Ellipse
2. Parabola
3. 5x2  3y2  30  0
3. Hyperbola
4. x 2  y 2  12x  4y  31  0
4. Circle
5.  x 2  y 2  6x  6y  4  0
5. Hyperbola
6. x 2  8x  4y  16  0
7. 3 x  3 y  30x  59  0
2
2
8. x  2 y  8 x  7  0
2
2
9. 4 x  y  16x  6 y  3  0
2
2
10. 3 x  y  4 y  3  0
2
2
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
B2 − 4AC < 0 & either B≠0 or A≠C
B2 − 4AC = 0
B2 − 4AC > 0
Circle
Ellipse
Parabola
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 Conic
2x2 + y2 −4x − 4 = 0
Complete the Square
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
2( x  1)
y

1
6
6
2
2
( x  1)
y

1
3
6
V(1±√6), CV(1±√3)
2
2
Steps to Complete the Square
1. 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
2
x  4 y  2x  8 y  1  0
2
1
2
2


x  2x  4( y  2 y)  1
 2 
2
2
x  2x  ___ 4( y  2 y  ___) 1  ___ ___
2
2
( x  2x  1)  4( y  2 y  1)  1  1  (4)(1)
2
2
( x 1)  4( y 1)  4
2
2
( x  1) 2 4( y  1) 2 4


4
4
4
2
2
( x  1) ( y  1)

1
4
1
What is the general 2nd degree equation for any
conic?
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
What information can the discriminant tell you
about a conic?
B2- 4AC < 0, B = 0, A = C
Circle
B2- 4AC < 0, B ≠ 0, A ≠ C
Ellipse
B2- 4AC = 0,
Parabola
B2- 4AC > 0
Hyperbola
Assignment 10.6
Page 628, 29-55 odd