ellipses and hyperbolas worksheet.ks-ia2

Transcription

ID: 1
Algebra 2
Name___________________________________
Ellipses and Hyperbolas
©q x27071w4w 2KLuTt1ah USVo7fdtkwYaQrpeB ALELyCH.E 6 9A1lLl6 brwijgKhjtdsD PrPeWsCerrsvYefdh.n
Identify the vertices, co-vertices, foci, length of the major axis, and length of the minor axis of each ellipse.
Then sketch the graph.
2
2
2
x
y
1)
+
=1
49 25
2
x
y
2)
+
=1
9
16
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
−2
−2
−4
−4
−6
−6
−8
−8
2
2
y
3) x +
=1
36
−4
6
8 x
2
4
6
8 x
2
y
−6
4
x
y
4)
+
=1
36 25
2
−8
2
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
−2
−2
−4
−4
−6
−6
−8
−8
©J J2f0g1J4d NKsultra0 PSboOfFtawqaJrHeZ 9L5LbC3.b J 8A1lilo 2r4iRgxhatbs3 irZeUsbeIrqvkemdk.M 1 1MJaLdiex vwViptQhB 9I7n8faiinJiZtTeH yAplagCeLbPr5aC m2j.R
Worksheet by Kuta Software LLC
Identify the vertices, co-vertices, and foci of each ellipse. Then sketch the graph.
5) 49 x 2 + 4 y 2 − 196 = 0
6) x 2 + 25 y 2 − 25 = 0
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
7) 16 x 2 + 25 y 2 − 400 = 0
−6
−4
8 x
4
6
8 x
y
8
8
6
6
4
4
2
2
−2
6
8) 9 x 2 + 4 y 2 − 36 = 0
y
−8
4
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
©X w2C0x1v4n YK8u4taas FSCoefpt2wNaHr7eR SL3L4C2.S 9 dArlBla 4roiVgThgtWsy TrfecskecrQvxe0dg.N P uMhaxdEeE pwMi8thhy gIungfviDnSirt5eo OAflIgje8bPrNaE z2o.N
Use the information provided to write the standard form equation of each ellipse.
9) Vertices: (10, 0), (−10, 0)
Co-vertices: (0, 5), (0, −5)
10) Vertices: (0, 9), (0, −9)
Co-vertices: (8, 0), (−8, 0)
11) Vertices: (0, 13), (0, −13)
Foci: (0, 5), (0, −5)
12) Vertices: (5, 0), (−5, 0)
Foci: (3, 0), (−3, 0)
13) Foci: (4, 0), (−4, 0)
Co-vertices: (0, 3), (0, −3)
14) Foci: (0, 3), (0, −3)
Co-vertices: (4, 0), (−4, 0)
15) Foci: (4, 0), (−4, 0)
Endpoints of major axis: (5, 0), (−5, 0)
16) Foci: (0, 12), (0, −12)
Endpoints of minor axis: (5, 0), (−5, 0)
©A m2c0Q1S4a 1Keuetta8 mS2oXfrtRwQanrzea CL1LHC9.O M fAylPlb Cr2iegphpt8st DrGecsQejrOv9eodR.g z hMjaDd5e0 vwziRtyhk AI2nGfZi1ntiRt3em VA5lNgjeObirOaZ 42F.Q
Identify the center, vertices, co-vertices, and foci of each. Then sketch the graph.
17)
( x − 1) 2
36
2
y
+
=1
49
18)
( x − 3) 2
+
( y − 2) 2
9
y
−8
−6
−4
2
19) ( x + 3) +
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
( y − 1) 2
2
=1
20)
36
−6
−4
6
8 x
4
6
8 x
2
y
8
8
6
6
4
4
2
2
−2
4
( y − 1)
x
+
=1
25
4
y
−8
=1
16
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
©Q M2H0u1S4r 0KWuCt9ax oScoSfxtcwBaSrTeh dLyLWC8.H O XAxlFld 8rHivgPh5t6sM fr6eysbeIrtvreKdc.7 a QMIaTd1eQ jwfiItehJ SImnGfHivnnivtTeB fAHlCgeeLbDrVah R2d.7
Write the equation of the ellipse in standard form by completing the squares. Identify the center, vertices,
co-vertices, and foci. Then sketch the graph.
21) 4 x 2 + 9 y 2 − 18 y − 135 = 0
22) 9 x 2 + 4 y 2 + 18 x − 8 y − 131 = 0
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
23) 16 x 2 + y 2 − 192 x + 6 y + 569 = 0
−6
−4
8 x
y
8
8
6
6
4
4
2
2
−2
6
24) 4 x 2 + 25 y 2 + 16 x − 200 y + 316 = 0
y
−8
4
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
©a p2V0Q1E42 4K1uBthaQ nSQoEfGtbwYaqrheU ELqLkC6.G 8 ZAXl6lq Xrkilg3hutKs7 2rXexs6ecryvLehd6.E l dM6aNd1ej twbiPtkhj jIgnJffiAnWiJtUeP gAulzgreebhrtau e2V.O
4
6
8 x
Use the information provided to write the standard form equation of each ellipse.
25) Vertices: (−6, 3), (−6, −11)
Co-vertices: (−5, −4), (−7, −4)
26) Vertices: (2, 1), (2, −21)
Co-vertices: (11, −10), (−7, −10)
27) Foci: (19, 0), (−5, 0)
Co-vertices: (7, 5), (7, −5)
28) Foci: (−7, −5), (−7, −11)
Co-vertices: (−3, −8), (−11, −8)
29) Foci: (1 + 6 3, 7), (1 − 6 3, 7)
Co-vertices: (1, 13), (1, 1)
30) Foci: (−10 + 7, 2), (−10 − 7, 2)
Co-vertices: (−10, 5), (−10, −1)
31) Foci: (−10, −9 + 3 7 ), (−10, −9 − 3 7 )
Endpoints of major axis: (−10, 3), (−10, −21)
32) Foci: (−6 + 2 3, 5), (−6 − 2 3, 5)
Endpoints of minor axis: (−6, 7), (−6, 3)
©p Q2K0I1C4b nKFu4tqaT aSJovfctWwBabrUeS LLtLGC5.g t 9Aelilm 8rOiBgmhFtLsU BriebsGevr8v9eZdb.d o kMhaVdxeL dwfiWtZhE 8INnLfXiWnqiIt8eX VAllMgOeybYrJac m2Q.v
Identify the vertices, foci, and asymptotes of each hyperbola. Then sketch the graph.
2
2
2
y
x
−
=1
33)
16
4
y
34) x −
=1
16
2
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
−2
−2
−4
−4
−6
−6
−8
−8
x
y
35)
−
=1
4
25
y
x
36)
−
=1
25
25
2
2
2
−6
−4
6
8 x
2
4
6
8 x
y
8
8
6
6
4
4
2
2
−2
4
2
y
−8
2
2
4
6
8 x
−8
−6
−4
−2
−2
−2
−4
−4
−6
−6
−8
−8
©L u2R0o1G4A kKBuwtSax GSEo4fctIwHaYr2eo zLrLeCr.i H iALldl3 HruiVgohHtMsc TrZeZsYexrEvte2dD.4 X RM6aadleU Jwti5tRhs MINnLf8i0nniAtJeC IAMlqgtezbjrpam u25.G
37) −4 x 2 + 25 y 2 − 100 = 0
38) x 2 − 16 y 2 − 16 = 0
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
39) 4 x 2 − 9 y 2 − 36 = 0
−6
−4
8 x
6
8 x
y
8
8
6
6
4
4
2
2
−2
6
40) −9 x 2 + 25 y 2 − 225 = 0
y
−8
4
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
©j l260S1B4g iKnuYt0ai 2SWovfntBwNaHrLe1 KLqLiCj.Z G WAWlAlb 6rBivghhktfs7 qr5exsMeJrBvyewd8.2 a GMfaGd9ew Hw1iOtmh3 tIVn2fvifnRiQtseE xAklCgwe5b0r5a1 02a.S
4
Use the information provided to write the standard form equation of each hyperbola.
41) Vertices: (3, 0), (−3, 0)
Foci: (5, 0), (−5, 0)
42) Vertices: (0, 12), (0, −12)
Foci: (0, 13), (0, −13)
43) Vertices: (0, 5), (0, −5)
Foci: (0, 74 ), (0, − 74 )
44) Vertices: (7, 0), (−7, 0)
Foci: ( 53, 0), (− 53, 0)
Identify the center, vertices, foci, and asymptotes of each hyperbola. Then sketch the graph.
45)
( y + 1) 2
2
−
16
x
=1
9
46) ( x + 2) 2 −
( y − 2) 2
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
=1
9
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
©a C2Q071j4n dKPuctRaX 6Sdo2fKttwcaAr0eN RLuLaC8.O Z QABlrlh SrwicgphztAsN UrXeasZe0riv1eFdn.B y BM3aOd1eW IwSijtEhn wIEntfDiUnhirt8eb gAvlNgaetbBrVaT j2z.U
4
6
8 x
47)
( y − 1) 2
−
( x + 1) 2
4
=1
48)
9
( x − 2) 2
4
2
− ( y + 3) = 1
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
Write the equation of the hyperbola in standard form by completing the squares. Identify the center,
vertices, foci, and asymptotes of each. Then sketch the graph.
49) − x 2 + 9 y 2 + 2 x + 72 y + 134 = 0
50) 4 x 2 − 25 y 2 − 150 y − 325 = 0
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
©j h2u0H1a45 7KduCtraD vSqojfrtMwoaXrHey BL1LXCv.6 9 zAXlUlG QrviogIhTtzs0 KrMepsKeVrPvMeWdt.S L gMQaJd0ek swsiltmh0 tI4nwf7iBnriHtmek kAolwg4e4bzr7aM e2z.f
4
6
8 x
51) −16 x 2 + 9 y 2 + 32 x + 18 y − 151 = 0
52) 9 x 2 − 4 y 2 − 18 x + 8 y − 31 = 0
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
Use the information provided to write the standard form equation of each hyperbola.
53) Vertices: (3, −6), (−9, −6)
Foci: (−3 + 61, −6), (−3 −
61, −6)
55) Vertices: (3, 15), (3, −15)
Foci: (3, 17), (3, −17)
©Z 02h0K1e4v YKsuWt5aQ WSOoRfwtVwEajrVeO 9LQLpCE.g S DASltl3 UryiJgShyttsm Jr9eHsAeQrOvmeydD.j 7 yMHa6dDeO 4wNiHtUhW RIDnPfzi5n2ict5e3 oAeltg8eAbfrEaP 826.e
54) Vertices: (−6, 9), (−6, −11)
Foci: (−6, −1 + 5 5 ), (−6, −1 − 5 5 )
56) Vertices: (1, −4), (−1, −4)
Foci: ( 170, −4), (− 170, −4)
Answers to Ellipses and Hyperbolas (ID: 1)
1)
2)
y
8
6
4
2
−8 −6 −4 −2
2
4
y
Vertices: (7, 0)
(−7, 0)
Co-vertices: (0, 5)
(0, −5)
Foci: (2 6 , 0)
(−2 6 , 0)
8 xMajor Axis: 14 units
Minor Axis: 10 units
6
−2
Vertices: (0, 4)
(0, −4)
Co-vertices: (3, 0)
(−3, 0)
Foci: (0, 7 )
(0, − 7 )
Minor Axis: 6 units
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
−4
−4
−6
−6
−8
−8
3)
4)
y
6
4
2
−8 −6 −4 −2
2
4
y
Vertices: (0, 6)
(0, −6)
Co-vertices: (1, 0)
(−1, 0)
Foci: (0, 35 )
(0, − 35 )
Minor Axis: 2 units
8
6
−2
Vertices: (6, 0)
(−6, 0)
Co-vertices: (0, 5)
(0, −5)
Foci: ( 11 , 0)
(− 11 , 0)
Minor Axis: 10 units
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
−4
−4
−6
−6
−8
−8
5)
6)
y
6
4
2
−8 −6 −4 −2
y
Vertices: (0, 7)
(0, −7)
Co-vertices: (2, 0)
(−2, 0)
Foci: (0, 3 5 )
(0, −3 5 )
8
2
4
6
6
4
2
8 x
−2
Vertices: (5, 0)
(−5, 0)
Co-vertices: (0, 1)
(0, −1)
Foci: (2 6 , 0)
(−2 6 , 0)
8
−8 −6 −4 −2
2
4
6
8 x
−2
−4
−4
−6
−6
−8
−8
7)
8)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
6
4
2
−8 −6 −4 −2
2
−4
−6
−6
−8
−8
2
2
y
6
4
2
−8 −6 −4 −2
2
−2
2
x
y
10)
+
=1
64 81
x2
y2
14)
+
=1
16 25
8
4
6
4
6
8 x
−2
−4
x
y
9)
+
=1
100 25
x2
y2
13)
+
=1
25
9
17)
Vertices: (0, 3)
(0, −3)
Co-vertices: (2, 0)
(−2, 0)
Foci: (0, 5 )
(0, − 5 )
8
8 x
−2
2
y
Vertices: (5, 0)
(−5, 0)
Co-vertices: (0, 4)
(0, −4)
Foci: (3, 0)
(−3, 0)
Center: (1, 0)
Vertices: (1, 7)
(1, −7)
Co-vertices: (7, 0)
(−5, 0)
Foci: (1, 13 )
(1, − 13 )
8 x
2
2
2
x
y
11)
+
=1
144 169
x2
y2
15)
+
=1
25
9
18)
y
8
6
4
2
−8 −6 −4 −2
2
−2
−4
−4
−6
−6
−8
−8
©M u2100124W CKDuitqad aSYoMfetKwQaErKei YLOLTCF.d h 1Amljld NrIiugBh0t4sr OrmejsheWr6vse5dd.x a YMzaVdteq FwniStihL FIQnXf0iZnKiwthe8 zANlqgNetb9r2ad w22.r
2
x
y
12)
+
=1
25 16
x2
y2
16)
+
=1
25 169
4
6
Center: (3, 2)
Vertices: (3, 6)
(3, −2)
Co-vertices: (6, 2)
(0, 2)
Foci: (3, 2 + 7 )
(3, 2 − 7 )
8 x
19)
20)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
y
Center: (−3, 1)
Vertices: (−3, 7)
(−3, −5)
Co-vertices: (−2, 1)
(−4, 1)
Foci: (−3, 1 + 35 )
(−3, 1 − 35 )
8 x
8
6
4
2
−8 −6 −4 −2
−4
−4
−6
−6
−8
−8
21)
4
6
22)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
y
Center: (0, 1)
Vertices: (6, 1)
(−6, 1)
Co-vertices: (0, 5)
(0, −3)
Foci: (2 5 , 1)
(−2 5 , 1)
8 x
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
−4
−4
−6
−6
−8
−8
23)
Center: (−1, 1)
Vertices: (−1, 7)
(−1, −5)
Co-vertices: (3, 1)
(−5, 1)
Foci: (−1, 1 + 2 5 )
(−1, 1 − 2 5 )
8 x
24)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
( x + 6) 2 +
y
Center: (6, −3)
Vertices: (6, 1)
(6, −7)
Co-vertices: (7, −3)
(5, −3)
Foci: (6, −3 + 15 )
(6, −3 − 15 )
8 x
8
6
4
2
−8 −6 −4 −2
2
−4
−6
−6
−8
−8
( y + 4) 2
y
( y + 10) 2
6
4
2
2
4
6
−2
6
( x − 7) 2
2
y
27)
+
=1
169
25
( x + 10) 2 ( y − 2) 2
30)
+
=1
16
9
Vertices: (1, 0)
(−1, 0)
Foci: ( 17 , 0)
(− 17 , 0)
Asym.: y = 4 x
y = −4 x
8
6
4
2
8 x
−8 −6 −4 −2
2
4
6
−2
−4
−4
−6
−6
−8
−8
35)
Center: (−2, 4)
Vertices: (3, 4)
(−7, 4)
Co-vertices: (−2, 6)
(−2, 2)
Foci: (−2 + 21 , 4)
(−2 − 21 , 4)
8 x
y
Vertices: (0, 4)
(0, −4)
Foci: (0, 2 5 )
(0, −2 5 )
Asym.: y = 2 x
y = −2 x
8
−8 −6 −4 −2
( x − 2) 2
+
=1
81
121
( x − 1) 2 ( y − 7) 2
29)
+
=1
144
36
( x + 6) 2 ( y − 5) 2
32)
+
=1
16
4
34)
26)
4
−2
−4
=1
49
( x + 7) 2 ( y + 8) 2
28)
+
=1
16
25
( x + 10) 2 ( y + 9) 2
31)
+
=1
81
144
33)
25)
2
−2
Center: (0, 1)
Vertices: (5, 1)
(−5, 1)
Co-vertices: (0, 3)
(0, −1)
Foci: ( 21 , 1)
(− 21 , 1)
8 x
8 x
36)
y
8
6
4
2
−8 −6 −4 −2
2
−2
4
6
Vertices: (2, 0)
(−2, 0)
Foci: ( 29 , 0)
(− 29 , 0)
5
Asym.: y = x
2
8 x
5
y=− x
2
y
6
4
2
−8 −6 −4 −2
2
−2
−4
−4
−6
−6
−8
−8
©x B2p0g1V44 lKMuCtQal KSaoffRtYwqa9r3eE fL9LpCn.5 F YARlnlw 3roisgWhztYsK PrrezsBeUrTvHeHdG.T 6 lMnakdxeo KwYiQtUh5 gIfnBfWienxiutbe1 aAwlYgieYb9rBaA T2E.5
Vertices: (0, 5)
(0, −5)
Foci: (0, 5 2 )
(0, −5 2 )
Asym.: y = x
y = −x
8
4
6
8 x
37)
38)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
−4
−4
−6
−6
−8
−8
39)
Vertices: (4, 0)
(−4, 0)
Foci: ( 17 , 0)
(− 17 , 0)
1
Asym.: y = x
4
8 x
1
y=− x
4
40)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
41)
y
Vertices: (0, 2)
(0, −2)
Foci: (0, 29 )
(0, − 29 )
2
Asym.: y = x
5
8 x
2
y=− x
5
y
Vertices: (3, 0)
(−3, 0)
Foci: ( 13 , 0)
(− 13 , 0)
2
Asym.: y = x
3
8 x
2
y=− x
3
8
6
4
2
−8 −6 −4 −2
2
−4
−4
−6
−6
−8
−8
x2
y2
−
=1
9
16
42)
y2
x2
−
=1
144
25
45)
4
6
−2
43)
Vertices: (0, 3)
(0, −3)
Foci: (0, 34 )
(0, − 34 )
3
Asym.: y = x
5
8 x
3
y=− x
5
y2
x2
−
=1
25
49
44)
x2
y2
−
=1
49
4
46)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
y
Vertices: (0, 3)
(0, −5)
Foci: (0, 4)
(0, −6)
4
Asym.: y = x − 1
3
8 x
4
y=− x−1
3
6
4
2
−8 −6 −4 −2
2
4
6
−2
−4
−4
−6
−6
−8
−8
47)
Vertices: (−1, 2)
(−3, 2)
Foci: (−2 + 10 , 2)
(−2 − 10 , 2)
Asym.: y = 3 x + 8
y = −3 x − 4
8
8 x
48)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
y
Vertices: (−1, 3)
(−1, −1)
Foci: (−1, 1 + 13 )
(−1, 1 − 13 )
2
5
Asym.: y = x +
3
3
8 x
2
1
y=− x+
3
3
8
6
4
2
−8 −6 −4 −2
2
−4
−4
−6
−6
−8
−8
49)
4
6
−2
Vertices: (4, −3)
(0, −3)
Foci: (2 + 5 , −3)
(2 − 5 , −3)
1
Asym.: y = x − 4
2
8 x
1
y=− x−2
2
50)
y
8
6
4
2
−8 −6 −4 −2
2
−2
4
6
Vertices: (1, −3)
(1, −5)
Foci: (1, −4 + 10 )
(1, −4 − 10 )
1
13
Asym.: y = x −
3
3
8 x
1
11
y=− x−
3
3
y
8
6
4
2
−8 −6 −4 −2
2
−2
−4
−4
−6
−6
−8
−8
©K l2q0x1H4z QKMuWtvaR 1SeoPfDtKwTaRrQev TLBLzC4.N V 7A1l8lN Brji0gehLtRsA 9rZevsDe3rZvYeCdt.W d iMTazdKeR lwtiytbhl gIXnhfeinnEi0tLeu dAqlWg3eCbHrbaP 02K.p
4
6
Vertices: (5, −3)
(−5, −3)
Foci: ( 29 , −3)
(− 29 , −3)
2
Asym.: y = x − 3
5
8 x
2
y=− x−3
5
51)
52)
y
8
6
4
2
−8 −6 −4 −2
2
4
6
−2
y
Vertices: (1, 3)
(1, −5)
Foci: (1, 4)
(1, −6)
4
7
Asym.: y = x −
3
3
8 x
4
1
y=− x+
3
3
8
6
4
2
−8 −6 −4 −2
−4
−4
−6
−6
−8
53)
( x + 3)
2
−
56) x −
4
6
−8
( y + 6)
36
2
2
−2
Vertices: (3, 1)
(−1, 1)
Foci: (1 + 13 , 1)
(1 − 13 , 1)
3
1
Asym.: y = x −
2
2
8 x
3
5
y=− x+
2
2
25
( y + 4) 2
2
=1
54)
( y + 1)
100
2
−
( x + 6)
25
2
=1
( x − 3) 2
y2
55)
−
=1
225
64
=1
169
©7 x2r081K4P IKDuZtQa0 cSQoRfVtIwVaJrseX 6LDLRCl.m B 5AClhlS 0ruiygjhbtDso Qrjeys4eDr8vZe1dM.v 4 XMyaodne6 LwtiztWhX CI5njfsiRntiMt1eJ rAXl2g6epbRrBaj c2Y.S

ellipses and hyperbolas worksheet.ks-ia2

Transcription

Similar documents