SOLUTION: PLEASE HELP!!! An ellipse has a center at (-4,6) and passes through (-4,9) and (2,6). Find the equation of the ellipse, the coordinates of the vertices and foci.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: PLEASE HELP!!! An ellipse has a center at (-4,6) and passes through (-4,9) and (2,6). Find the equation of the ellipse, the coordinates of the vertices and foci.      Log On


   

Question 820174: PLEASE HELP!!! An ellipse has a center at (-4,6) and passes through (-4,9) and (2,6). Find the equation of the ellipse, the coordinates of the vertices and foci.
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
An ellipse has a center at (-4,6) and passes through (-4,9) and (2,6). Find the equation of the ellipse, the coordinates of the vertices and foci.
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Given coordinates show this is an ellipse with horizontal major axis.
Its standard form: %28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1,a>b, (h,k)=(x,y) coordinates of center.
..
%28x%2B4%29%5E2%2Fa%5E2%2B%28y-6%29%5E2%2Fb%5E2=1
plug in coordinates of one given point (-4,9) on the ellipse.
%28-4%2B4%29%5E2%2Fa%5E2%2B%289-6%29%5E2%2Fb%5E2=1
0%2B%283%29%5E2%2Fb%5E2=1
b^2=9
b=3
..
plug in coordinates of 2nd given point (2,6) on the ellipse.
%282%2B4%29%5E2%2Fa%5E2%2B%286-6%29%5E2%2Fb%5E2=1
%282%2B4%29%5E2%2Fa%5E2%2B0=1
%286%29%5E2%2Fa%5E2%2B0=1
a^2=36
a=6
..
equation of given ellipse:
%28x%2B4%29%5E2%2F36%2B%28y-6%29%5E2%2F9=1
..
vertices:(-4±a,6)=(4±6,6)=(-2,6) and (10,6)
c^2=a^2-b^2=36-9=27
c=√27≈5.2
..
foci:(-4±c,6)=(4±5.2,6)=(-1.2,6) and (9.2,6)