SOLUTION: Find the vertices and foci of the ellipse given by: x^2+9y^2-24x+18y+9=0

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Question 97784: Find the vertices and foci of the ellipse given by:
x^2+9y^2-24x+18y+9=0
Answer by mathslover(157) About Me  (Show Source):
You can put this solution on YOUR website!
The standard form of the equation of an ellipse is
+%28x-h%29%5E2%2Fa%5E2++%2B+%28y-k%29%5E2%2Fb%5E2+=1+ where (h,k) is the center of the ellipse
we first try and put the given equation in the standard form
x%5E2%2B9y%5E2-24x%2B18y%2B9=0
+x%5E2+-24x+%2B+144++%2B+9y%5E2+%2B+18y+%2B+9+-144+=0
+x%5E2+-24x+%2B+144++%2B+9%28y%5E2+%2B+2y+%2B+1%29+=144
%28x-12%29%5E2+%2B+9%28y+%2B+1%29%5E2+=144+
%28%28x-12%29%5E2%29%2F144+%2B+%289%28y+%2B+1%29%5E2%29%2F144+=1+
%28%28x-12%29%5E2%29%2F144+%2B+%28y+%2B+1%29%5E2%2F16+=1+
%28%28x-12%29%5E2%29%2F12%5E2+%2B+%28y+-+%28-1%29%29%5E2%2F4%5E2+=1+
so we have the center (h,k) as (12,-1)
also a=12 and b=4
Since a > b the major axis is horizontal
vertices are given by (h-a,k) and (h+a,k)
so we have ( 12-12,-1) and (12+12,-1)
vertices are (0,-1) and (24,-1)
focii are at points given by (h-c,k) and (h+c,k) where c = sqrt%28a%5E2+-b%5E2%29
(12- 8*sqrt2,-1) and (12+ 8*sqrt2,-1)