SOLUTION: Find the center, foci, vertices, length of major axis, and length of minor axis of the ellipse 25(x-2)^2 + 4(y+5)^2 = 100. Sketch the graph of the ellipse. Thanks in advance... (^2

The standard forms for ellipses are

 for ellipses like this  where a > b

 for ellipses like this  where a < b



To get 1 on the right, divide through by 



Simplify:



Since the larger number is under the expression in y the ellipse is 
of the form  
and is like this  

, ,  or , , or 

center = (h,k) = (2,-5)

So let's begin by plotting the center (2,-5)

  

Next we draw a vertical line beginning at the center
(2,-5) and going upward  units, which is one-half the
major axis.  This ends in the point (2,0) which is the upper
vertex. 
 





Next we draw a vertical line beginning at the center
(2,-5) and going downward  units, which is one-half the
major axis.  This ends in the point (2,-10) which is the lower
vertex.  That green line is the major axis, and it is 10 units long.
 
 


Next we draw a horizontal line beginning at the center
(2,-5) and going to the right  units, which is one-half the
minor axis. This ends in the point (4,-5) which is the right co-vertex. 
 


Next we draw a horizontal line beginning at the center
(2,-5) and going to the leftt  units, which is one-half the
minor axis. This ends in the point (0,-5) which is the left co-vertex.
The horizontal green line is is the minor axis, and it is 4 units long.
 


Now we can sketch in the ellipse:



Finally we find the foci.  They are the two points on the major axis
w2hich are c units from the center, where c is calculated by







So we add  to the y-coordinate of the center
to find the upper focus, which is the point
(2,), which is about (2,9.6),
marked in red below.


 
And we subtract  from the y-coordinate of the center
to find the lower focus, which is the point
(2,), which is about (2,0.4),
also marked in red below.



Edwin