Question 144052
The center of the general ellipse {{{(x-h)^2/a^2 + (y-k)^2/b^2 = 1}}} is (h,k). 



Since h=3 and k=-1, the center is (3,-1)



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Now we need to find the values of "a" and "b". So {{{a^2=4}}} ===>{{{a=2}}} and {{{b^2=9}}} ===>{{{b=3}}}


To find the vertices, simply start with the center (3,-1)


Add the value {{{a=2}}} to the x-coordinate of the center to get (3+2,-1)--->(5,-1) This is the right-most vertex


Subtract the value {{{a=2}}} from the x-coordinate of the center to get (3-2,-1)--->(1,-1) This is the left-most vertex



Add the value {{{b=3}}} to the y-coordinate of the center to get (3,-1+3)--->(3,2) This is the top-most vertex


Subtract the value {{{b=3}}} from the y-coordinate of the center to get (3,-1-3)--->(3,-4) This is the bottom-most vertex



So the vertices are (5,-1), (1,-1), (3,2), and (3,-4)




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To find the vertices, we must find the distance from the center to each of the foci


So use this formula to find the distance

{{{c^2=b^2-a^2}}} where c is the distance from the center to the focus



{{{c^2=3^2-2^2}}} Plug in {{{a=2}}} and {{{b=3}}}



{{{c^2=9-4}}} Square each term



{{{c^2=5}}} Subtract



{{{c=sqrt(5)}}} Take the square root of both sides



Now add this value {{{c=sqrt(5)}}} to the y-coordinate of the center (3,-1) to get *[Tex \LARGE \left(3,-1+\sqrt{5}\right)]. This is the top-most focus.


Now subtract this value {{{c=sqrt(5)}}} from the y-coordinate of the center (3,-1) to get *[Tex \LARGE \left(3,-1-\sqrt{5}\right)]. This is the bottom-most focus.


So the foci are *[Tex \LARGE \left(3,-1+\sqrt{5}\right)] and *[Tex \LARGE \left(3,-1-\sqrt{5}\right)]






Here's a picture to verify the answers.


{{{drawing(500,500,-10,10,-10,10,
graph(500,500,-10,10,-10,10,0),
ellipse( 3, -1, 4, 6 ),
green(circle(5,-1,0.08)),
green(circle(5,-1,0.1)),
green(circle(5,-1,0.12)),
green(circle(1,-1,0.08)),
green(circle(1,-1,0.1)),
green(circle(1,-1,0.12)),
green(circle(3,2,0.08)),
green(circle(3,2,0.1)),
green(circle(3,2,0.12)),
green(circle(3,-4,0.08)),
green(circle(3,-4,0.1)),
green(circle(3,-4,0.12)),
red(circle(3,1.23606,0.08)),
red(circle(3,1.23606,0.1)),
red(circle(3,1.23606,0.12)),
red(circle(3,-3.23606,0.08)),
red(circle(3,-3.23606,0.1)),
red(circle(3,-3.23606,0.12))
)}}} Graph of {{{(x-3)^2/4 + (y+1)^2/9 = 1}}} with the vertices (green) and the foci (red)