SOLUTION: write the equation of the ellipse in standard form having the given properties. center(4,6) vertex(9,6)(0,8) is on the ellipse

Plot those points



Sketch in the ellipse approximately:



Since the ellipse is wider than it is tall, it has the 
standard equation

 = 1

And since the center is (h,k) = (4,6), we have

 = 1

Now we draw in the semi-major axis, which connects the
center to the vertex:



We can observe that the semi-major axis is 5 units long, so
we know that a = 5, so we can substitute that and our
equation so far is:

 = 1

or

 = 1

Now since we know that the ellipse contains the point
(x,y) = (0,8), we substitute

 = 1

 = 1

 = 1

Clear of fractions by multiplying through by LCD = 









Since b is the length of the semi-minor axis,
we can now draw in the semi-minor axis 10/3 or
3 units long, which goes from the 
center up to the ellipse:



To draw the ellipse more accurately we can now sketch
in the complete major and minor axes:



To finish the equation, we substitute  for 

 = 1 

and that is the desired equation in standard form.

Edwin