1.
a)
C(2,3),2a=8,2b=4
(8/2)2(x−2)2+(4/2)2(y−3)2=1
16(x−2)2+4(y−3)2=1 b)
C(1,−2),2a=8,ε=ac=aa2−b2=23
a2−b2=43a2
b=2a=2
16(x−1)2+4(y+2)2=1
2.
a)
F1(−2,1),F2(4,1),2a=10
a=5
k=1,h−c=−2,h+c=4 C(1,1)
c=a2−b2=3
b=4
25(x−1)2+16(y−1)2=1
b)
F1(−3,0),F2(−3,4),2b=6
b=3
h=−3,k−c=0,k+c=4 C(−3,2)
c=a2−b2=2
a=13
9(x+3)2+13(y−2)2=1
3.
a)
F1(−2,2),F2(4,2),ac=53
k=2,h−c=−2,h+c=4 C(1,2)
c=a2−b2=3
a=3(53)=5,b=52−32=4
25(x−1)2+16(y−2)2=1
b)
F(±2,0),x=±8
k=0,
h+ae=2,h−ae=−2
h+ea=8,h−ea=−8
h=0
a=8e,8e2=2
e=ac=21,a=4,c=a2−b2=2 C(0,0)
b=23
16x2+12y2=1
4.
a)
F1(0,0)
V1(5,0)
e=0.5
k=0,
h+ae=0,h+a=5
h+0.5a=0,h+a=5
a=10,h=−5 or
h−ae=0,h+a=5
h−0.5a=0,h+a=5
a=310,h=35
a=10,h=−5 C(−5,0)
c=a2−b2=5
b=53
100(x+5)2+75y2=1
a=310,h=35
C(35,0)
c=a2−b2=35
b=353
100/9(x−35)2+75/9y2=1
b)
F1(0,0)
V1(0,2)
e=0.6
h=0,k+ae=0,k+a=2
k+0.6a=0,k+a=2
a=5,k=−3 or
k−ae=0,k+a=2
k−0.6a=0,k+a=2
a=45,k=43
a=5,k=−3 C(0,−3)
c=a2−b2=3
b=4
16x2+25(y+3)2=1
a=45,k=43
C(0,43)
c=a2−b2=43
b=1
1x2+25/16(y−43)2=1
5.
a)
Center (1,3)
Vertices (1,−1)
b2(x−h)2+a2(y−k)2=1Center (h,k)
Vertices (h,k±a)
h=1,k=3
3−a=−1=>a=4 Passing through the origin
b2(0−1)2+42(0−3)2=1
b2=16/7
16/7(x−1)2+16(y−3)2=1
b)
Center (1,1)
Vertices (3,1)
a2(x−h)2+b2(y−k)2=1Center (h,k)
Vertices (h±a,k)
h=1,k=1
1+a=3=>a=2 Passing through the origin
22(0−1)2+b2(0−1)2=1
b2=4/3
4(x−1)2+4/3(y−1)2=1
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