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Question 975420: Given the ellipse: 9x2 + 4y2 + 18x - 48y + 117 = 0
a) Put the equation in standard form
b) Horizontal or vertical major axis?
c) Find center, vertices, foci,
length of major and minor axes,
and eccentricity
d) Sketch a graph of this ellipse.
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! 9x2 + 4y2 + 18x - 48y + 117 = 0
Rewrite as
9x^2 + 18x + ;;;; +4y^2 - 48 y =-117;; complete first square: 9(x^2+2x+1);; second 4(y^2-12y+36)
9(x^2+2x+1)+ 4(y^2-12y+36)=-117 + 9 +144 ;;remember to multiply the coefficient by the constant that was factored out
9(x^2+2x+1)+ 4(y^2-12y+36)=36 ;;must divide both sides by 36 to make the right side 1.
(x+1)^2/4 + (y-6)^2/9=1
Center is at (-1,6)
Major axis is along y-axis. And it is a^2=9; a=3 Vertices are at (-1,3) and (-1,9)
Minor axis is along x-axis. And it is b^2-4; b=2 These are at (1,6) and (-3,6)
Foci are a^2-c^2=b^2 They are at sqrt (5) from center. (-1, 6+ sqrt (5)) and (-1, 6-sqrt (5))
Eccentricity is c/a which is sqrt(5)/3
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