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Alg 3/Trig (10) 1
Conics
Parabolas Geometer Sketchpad
2y = a(x h) + k ?
Def: The set of all points in a plane equidistant from a point (focus) and a line (directrix).
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2(x -h) =4p(y -k)
2(y k) =4p(x h)
(h,k)
p p
F
4p
If p<0
If p < 0
If p > 0
p
p
4p
= (h,k)
p>0
Alg 3/Trig (10) 2
Conics
EXAMPLES
1. 2x 4y =
2. 2y 4x =
3. 2x 2x 8y 7 = 0
4. 22y y x 8 0 + + =
Alg 3/Trig (10) 3
Conics
Algebra 3 Assignment # 1
Sketch a graph of each of the following. Label the vertex, focus, and directrix please.
(1) 2x 2x 4y 9 0
(2) 2y 4y 8x 20 0
(3) 2x 2x 6y 19 0
(4) 2y 8x 32 0
(5) 24x 4x 12y 17 0
(6) 2y 6y 6x 21 0
(7) 2x 4x 4y 4 0
(8) 24y 12y 20x 19 0
Alg 3/Trig (10) 4
Conics
Algebra 3 Assignment # 1 Answers
(1) 2
x 1 4 y 2
(2) 2
y 2 8 x 3
(3) 2
x 1 6 y 3
(4) 2y 8 x 4
(5) 2
312 2
x 3 y
(6) 2
y 3 6 x 2
(7) 2
x 2 4 y 0
(8) 2
3 12 2
y 5 x
Alg 3/Trig (10) 5
Conics
Parabolas Day 2
1. Find the equation of the parabola with focus at (1,5) and directrix at y = 9.
2. Write the equation of the parabola with the vertex at (3, 5 ) and focus at ( 3, 7 ).
Alg 3/Trig (10) 6
Conics
3) Focus at ( 3, -1 ) and directrix at x = 1.
Alg 3/Trig (10) 7
Conics
Algebra 3 Assignment # 2
Write the equation of a parabola which satisfies each of the following please
(1) focus at (0 , 1) , directrix is y = 1.
(2) vertex at ( 2 , 1) , focus at ( 2 , 5).
(3) focus at (4 , 2) , directrix is x = 6.
(4) vertex at (1 , 3) , directrix is x = 5.
(5) vertex at 0 , 21
, focus at , 41
21
.
(6) directrix is x = 0 , focus at , 1 25
.
(7) focus at (0 , 0) , directrix is x = 5.
(8) vertex at (2 , 2) , passes through (0 , 0).
Alg 3/Trig (10) 8
Conics
Algebra 3 Assignment # 2 Answers
(1) 4y x2
(2) 1 y 16 2 x 2
(3) 5 x 4 2 y 2
(4) 1 x 16 3 y 2
(5) 2
12
x y
(6) x 2 y 212
25
(7) x 10 y252
(8) 2 y 2 2 x 2 or 2 x 2 2 y 2
Alg 3/Trig (10) 9
Conics
ELLIPSE DEF OF ELLIPSE
The ellipse is the set of all points in a plane such that the sum of the distances from two
fixed points, (called the foci) is a constant.
PF 1 + PF 2 = 2a 2 2
2 2
x-h y-k+ =1
a b
Major axis =
Minor axis =
2 2 2c a b
P(x,y)
F 1 (-c,0)
V(a,0)
F 2 (c,0)
V(-a,0)
(0,b)
(0,-b)
2 2
2 2
x-h y-k+ =1
b a
Alg 3/Trig (10) 10
Conics
ELLIPSES
(h, k) (h + a, k) (h-a, k)
(h, k+b)
(h+c, k)
(h, k-b)
(h-c, k)
(h, k) (h + b, k)
(h, k-a)
(h, k+a)
(h, k+c )
(h, k-c)
(h, k-b)
2 2
2 2
1- -y k
b
x
a
h
22
2 2
1y k
a
x
b
h
Alg 3/Trig (10) 11
Conics
Sketch the graph of the following ellipses. Find the coordinates of the vertices and the
foci.
1. 2 220x + y = 100
2. 2 2x 10x 4y 16y = 25 + +
3. 2 2 + = 16x y 16
4. 2 2x 4x 9y = 5
Alg 3/Trig (10) 12
Conics
5. 2 24x + 24x + 13y - 26y = 3
Alg 3/Trig (10) 13
Conics
Algebra 3 Assignment # 3
Sketch a graph of each of the following please. Label the center, endpoints of the major and minor
axes, and the focus points.
(1) 400 16y x25 22 (5) 0 59 16y 50x 4y x25 22
(2) 144 16y x9 22 (6) 0 9 8y 6x 4y x 22
(3) 4 y x4 22 (7) 0 12 4y y x4 22
(4) 0 61 54y 16x 9y x4 22 (8) 0 19 16y 8x 4y x4 22
Alg 3/Trig (10) 14
Conics
Algebra 3 Assignment # 3 Answers
(1)
2 2x y 1
16 25 (5)
2 2x 1 y 2
1 4 25
(2)
2 2x y 1
16 9 (6)
2 2x 3 y 1
1 4 1
(3)
2 2x y 1
1 4 (7)
2 2x 0 y 2
1 4 16
(4)
2 2x 2 y 3
1 9 4
(8) 2 2 1
4x 1 y 2
Alg 3/Trig (10) 15
Conics
Ellipse Day 2
TO FIND:
CENTER:
TO FIND “a”:
TO FIND “b”:
ORIENTATION:
1. Write the equation of the ellipse with
Center (1,1); Focus (1,3); Vertex (1,-9)
2. Foci (4,2) and (8,2) ; Major axis (MA) = (3,2), (9,2)
2 2
2 2
1- -y k
b
x
a
h 22
2 2
1y k
a
x
b
h
Alg 3/Trig (10) 16
Conics
3. MA = (3,2) and (9,2); c = 3 (what is “c” ?)
4. Write the equation if the foci are (-8, 1), (8, 1) and the minor axis is 6.
5. Foci at ( 7, 0 ), ( -7, 0 ), and vertices at (8, 0 ) and ( -8, 0 )
Alg 3/Trig (10) 17
Conics
Algebra 3 Assignment # 4
Write the equation of the ellipse which satisfies each of the following please.
(1) Foci at ( 2 , 3) and (4 , 3) if the length of the major axis is 10.
(2) Foci at ( 2 , 5) and ( 2 , 1) if the length of the major axis is 8.
(3) Foci at (0 , 3) and (4 , 3) , vertices at ( 4 , 3) and (8 , 3).
(4) Foci at ( 2 , 3) and ( 2 , 1) , vertices at ( 2 , 6) and ( 2 , 4).
(5) The endpoints of the major axis are ( 4 , 5) and (2 , 5) , the endpoints of the minor axis are ( 1 ,
7) and ( 1 , 3).
(6) Foci at ( 3 , 1) and ( 3 , 5) if the length of the minor axis is 6.
Alg 3/Trig (10) 18
Conics
Algebra 3 Assignment # 4 Answers
(1)
2 2x 1 y 3
1 25 16
(2)
2 2x 2 y 3
1 12 16
(3)
2 2x 2 y 3
1 36 32
(4)
2 2x 2 y 1
1 21 25
(5)
2 2x 1 y 5
1 9 4
(6)
2 2x 3 y 2
1 9 18
Alg 3/Trig (10) 19
Conics
The hyperbola is the set of all points in a plane such that the difference of the distances from two fixed
points, called foci, is constant.
2 2
2 2
y k x h = 1
a b
Hyperbolas
2 2
2 2
x h y k = 1
a b
2 2 2 = a + bc
Alg 3/Trig (10) 20
Conics
P(x,y)
(a,0) (-c,0)
(c,0) (-a,0)
(0,b)
(0,-b)
HYPERBOLAS
(h,k+a)
(h,k+c)
(h,k-a)
(h,k-c)
(h-b,k) (h+b,k)
(h,k)
2 2
2 21
( ) ( )x h y k
a b
2 2
2 21
( ) ( )y k x h
a b
(h,k)
(h,k+b)
(h-a,k) (h+a,k) (h-c,k) (h+c,k)
(h,k-b)
a -a
b
bm= x
a
bm= x
a
am= x
b
am= x
b
Alg 3/Trig (10) 21
Conics
Graph
`
1. 2 210x 10y = 40
2. 2 2y 6y 4x + 8x = 11
Alg 3/Trig (10) 22
Conics
Graph
3. 2 2y 6y x + 24x = 23
Alg 3/Trig (10) 23
Conics
4. 2 25x 10x 4y 16y = 31
Alg 3/Trig (10) 24
Conics
Algebra 3 Assignment # 5
Sketch a graph of each of the following please. Label the center, foci and asymptotes.
(1) 2 29x 4y 36
(2) 2 216y 9x 144
(3) 2 24x 25y 8x 100y 196 0
(4) 2 2y 4x 6y 40x 95 0
(5) 2 2x y 6x 4y 1 0
(6) 2 29y 16x 54y 32x 79 0
(7) 2 225x 9y 200x 175 0
(8) 2 2x 9y 4x 72y 149 0
Alg 3/Trig (10) 25
Conics
Algebra 3 Assignment # 5 Answers
(1)
2 2x y 1
4 9
(2) 1 16
x
9
y 22
(3)
2 2x 1 y 2
125 4
(4)
2 2y 3 x 5
14 1
(5)
2 2x 3 y 2
14 4
(6)
2 2y 3 x 1
116 9
(7)
2 2x 4 y 0
19 25
(8)
2 2x 2 y 4
19 1
Alg 3/Trig (10) 26
Conics
HYPERBOLAS DAY 2 WRITING EQUATIONS
TO FIND:
CENTER:
TO FIND “a”:
TO FIND “b”:
ORIENTATION:
1. Center = (1,3); Transverse axis endpoint = (1,7); Focus = (1,-2)
2. TA endpoints = (3,-3), (-5,-3); slope of asymptotes 7
2
Alg 3/Trig (10) 27
Conics
3. Foci = (1,0), (31,0); slope of asymptotes 4
3
4. Foci at (0, 4), vertices at (0, 2)
5. Vertices are ( -1, 3 ) and ( 5, 3 ) one focus is ( 7, 3 )
Alg 3/Trig (10) 28
Conics
Algebra 3 Assignment # 6
Write the equation of the hyperbola which satisfies each of the following please.
(1) Foci are ( 3 , 2) and (1 , 2) , the length of the transverse axis is 2
(2) Vertices are ( 1 , 3) and ( 1 , 1) , one focus is ( 1 , 5)
(3) Vertices are ( 6 , 2) and (0 , 2) , one focus is (2 , 2)
(4) Vertices are ( 2 , 3) and (4 , 3) , slopes of the asymptotes are 32
(5) Foci are (1 , 8) and (1 , 2) , one vertex is (1 , 5)
(6) Vertices are ( 4 , 2) and ( 4 , 2) , the length of the conjugate axis is 2
Alg 3/Trig (10) 29
Conics
Algebra 3 Assignment # 6 Answers
(1)
2 2x 1 y 2
11 3
(2)
2 2y 1 x 1
14 12
(3)
2 2x 3 y 2
19 16
(4)
2 2x 1 y 3
19 4
(5)
2 2y 3 x 1
14 21
(6)
22 x 4y 1
4 1
Alg 3/Trig (10) 30
Conics
Algebra 3Assignment # 7 ─ Review Worksheet
(1) Sketch a graph of each of the following. Label the center, vertices, foci, and asymptotes where
appropriate.
(a) 2x 4x 6y 20 0 (e)
2 29y x 18y 0
(b) 24y 4y 16x 47 0 (f)
2 29x 4y 54x 16y 29 0
(c) 2 216y + x 2x 32y + 13 = 0 (g)
2 216x + y 96x + 4y + 132 = 0
(d) 2 24x 9y 16x 18y 11 = 0 (h)
2 22x + 2y + 2x 6y 3 = 0
(2) Write the equation of a circle if the endpoints of a diameter are ( 5 , 2) , and (1 , 6)
(3) Write the equation of the parabola whose focus is ( 2 , 5), and whose directrix is x = 4.
(4) Write the equation of an ellipse if the endpoints of the major axis are (2 , 2) and (2 , 8), and one
of the focus points is at (2 , 0)
(5) Write the equation of the hyperbola whose foci are ( 2 , 5) and ( 2 , 3) if the length of the transverse
axis is 4.
(6) Write the equation of the parabola whose vertex is ( 3 , 4), and whose focus is (1 , 4).
(7) Write the equation of an ellipse if the center is (3 , 2), if the graph of the ellipse is to be tangent to the
coordinate axes.
(8) Write the equation of the hyperbola whose vertices are (7 , 3) and (1 , 3) if the slopes of its asymptotes
are 34 .
Alg 3/Trig (10) 31
Conics
Algebra 3Assignment # 7 ─ Review Worksheet Answers
(1) (a) 2
x 2 6 y 4 (e)
2 2y 1 x 1
1 9
(b) 2
12
y 4 x 3 (f)
2 2x 3 y 2
14 9
(c)
2 2x 1 y 1
14 (1/ 4)
(g)
2 2x 3 y + 2
+ 11 16
(d)
2 2x 2 y 1
+ 19 4
(h) 22
312 2
x + + y = 4
(2) 2 2
x + 2 + y 2 = 25
(3) 1 x12 5 y 2
(4)
2 2x 2 y + 3
+ 116 25
(5)
2 2y 1 x 2
14 12
(6) 2
y 4 16 x 3
(7)
2 2x 3 y 2
+ 19 4
(8)
2 2x 4 y 3
19 16