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Question 428547: Can someone show me how to do these two problems please?
1:Write an equation for the hyperbola where all points on the hyperbola are 68 units closer to one focus than the other. The foci are located at (0,0) and (250,0).
2: Write an equation for an ellipse with center (1,-3), vertices (1,2) and (1,-8) and co-vertices (4,-3) and (-2,-3)
Show me the steps please, I need to learn how to do this.
Thanks
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! 1:Write an equation for the hyperbola where all points on the hyperbola are 68 units closer to one focus than the other. The foci are located at (0,0) and (250,0).
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Standard form of hyperbola with horizontal transverse axis(opens sideways):
(x-h)^2a^2-(y-k)^2/b^2=1
Standard form of hyperbola with Vertical transverse axis(opens up and down):
(y-k)^2a^2-(x-h)^2/b^2=1
(h,k)=(xy) coordinates of the center for both forms
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2a=68 (points on hyperbola closer to one focus than the other)
a=34
a^2=1156
Center:(125,0)
Foci:(0,0) and (250,0)=c (given location of foci)
Tlhis shows transfer axis is horizontal at y=0
c=125
c^2=15625
c^2=a^2+b^2
b^2=c^2-a^2=15625-1156=14469
b=sqrt(14469)=120.29
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Equation of Hyperbola:
(x-125)^2/1156-y^2/14469=1
Equations of asymptotes: (use standard form for a straight line, y=mx+b, with slopes +-b/a and a point on the line which is the center,(125.0)
y=(120.29/34)x-442
y=-(120.29/34)x+442
see the graph below which can serve as a check on the answer
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y=(((((x-125)^2)/1156)-1)*14469)^.5
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2: Write an equation for an ellipse with center (1,-3), vertices (1,2) and (1,-8) and co-vertices (4,-3) and (-2,-3)
Standard form of ellipse for horizontal major axis: (x-h)^2/a^2+(y-k)^2/b^2=1 (a>b)
Standard form of ellipse for vertical major axis: (y-k)^2/a^2+(x-h)^2/b^2=1 (a>b)
In both forms, (h,k) represent the (x,y) coordinates of the center.
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Given data shows ellipse has a vertical major axis on the line,x=1, so it is of the second form described above.
Length of major axis =10=2a
a=5
a^2=25
Length of minor axis=6=2b
b=3
b^2=9
center:(1,-3) given
We now have all the information we need to write an equation for the ellipse.
(y+3)^2/25+(x-1)^2/9=1
see the graph below to check the answers
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y=(25(1-(x-1)^2/9))^.5-3
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