Transcript Conics (Chapter 6 Short Subject)

• CIRCLES
General Equation: (x – h)2 + (y – k)2 = r2
where (h, k) is the center and r is the radius
Graph the following equation:
(x + 2)2 + y2 = 25
Note:
Addition
and Same
Coefficients
center
(-2,0)
• ELLIPSES
General Equation: (x – h)2 + (y – k)2 = 1
a2
b2
where (h, k) is the center.
The vertices are a units right and left from
(h, k) and b units up and down from (h, k)
Center:
(h, k)
Note:
Addition and
Different
Coefficients
b
a
Major axis: 2b
Minor axis: 2a
a
b
since 0 < a < b (ellipse is vertical)
Graph the following equation:
(x – 7)2 + (y + 2)2 = 1
25
16
Center: (7, -2)
Since a2 = 25, then a =  5
Major axis
Since b2 = 16, then b =  4
Minor axis
Major:
(7 + 5, -2)  (12, -2)
(7  5, -2)  (2, -2)
Minor:
(7, -2 + 4)  (7, 2)
(7, -2  4)  (7, -6)
Graph the following equation:
4x2 + 25y2 = 400
400
400
400
x2 + y2
100
16
=
1
Center: (0, 0)
Since a2 = 100, then a =  10
Major axis
Since b2 = 16, then b =  4
Minor axis
Major:
(0 + 10, 0)  (10, 0)
(0  10, 0)  (-10, 0)
Minor:
(0, 0 + 4)  (0, 4)
(0, 0  4)  (0, -4)
• HYPERBOLAS
I. General Equation: (x – h)2 – (y – k)2 = 1
a2
b2
where (h, k) is the center.
The vertices are (h + a, k) and (h – a, k) and
opens left and right (transverse axis is
horizontal).
Note:
Subtraction
and (+) xcoefficient
(h – a, k)
(h + a, k)
m of asymptotes =  b/a
• HYPERBOLAS
II. General Equation: (y – k)2 – (x – h)2 = 1
b2
a2
where (h, k) is the center.
The vertices are (h, k + b) and (h, k – b) and
opens up and down (transverse axis is
vertical).
Note:
Subtraction
and (+) y coefficient
m of asymptotes =  b/a
Graph the following equation:
25x2 – 4y2 = 400
400 400
400
x2
16
–
y2 =
100
Center: (0, 0)
Since a2 = 16, then a =  4
Since b2 = 100, then b =  10
Vertices: (0 + 4, 0)  (4, 0)
(0  4, 0)  (-4, 0)
1
m =  10/4 =  5/2
Vertices
(since it
opens left
and right)
• Putting Equations in Standard Form by
Completing the Square
Graph the following equation:
4y2 + 9x2 – 24y – 72x + 144 = 0
Group the x
terms and y
terms
9x2 – 72x + 4y2 – 24y = - 144
Factor and leave blanks to complete the square:
9(x2 – 8x + 16 ) + 4(y2 – 6y + 9 ) = - 144 + 144 + 36
Complete the square and fill in blanks
9(x – 4)2 + 4(y – 3)2 = 36
Ellipse
Graph the following equation:
4y2 + 9x2 – 24y – 72x + 144 = 0
9(x – 4)2 + 4(y – 3)2 = 36
36
36
36
(x – 4)2 + (y – 3)2 = 1
Center: (4, 3)
4
9
Since a2 = 4, then a =  2
Minor axis
Since b2 = 9, then b =  3
Major axis
Minor:
(4 + 2, 3)  (6, 3)
(4  2, 3)  (2, 3)
Major:
(4, 3 + 3)  (4, 6)
(4, 3  3)  (4, 0)
Graph the following equation:
12y2 – 4x2 + 72y + 16x + 44 = 0
12y2 + 72y – 4x2 + 16x = - 44
12(y2 + 6y + 9 ) – 4(x2 – 4x + 4 ) = - 44 + 108 + - 16 )
12(y + 3)2 – 4(x – 2)2 = 48
(y + 3)2 – (x – 2)2 = 1
Hyperbola
– opens up
& down
Center: (2, -3)
4
12
Since b2 = 4, then b =  2
Since a2 = 12, then a =  2 3   3.5
Vertices: (2, -3 + 2)  (2, -1)
(2, -3 – 2)  (2, -5)
Vertices (since it
opens up & down)