SOLUTION: Hi, I'm supposed to write the equation of an ellipse with foci at (4,1) and (4,-7) and vertices at (4,2) and (4,-8) but I can't figure out what to do after the plus sign. I know t

To do ellipse problems, you must ALWAYS draw the graph.

You must also know all the parts of an ellipse, and the
equations and formulas.

They are:
 
1. The center, which has coordinates (h,k).
2. The major axis = the longesr chord that passes through the center,
bisected at the center. Its endpoints are the two vertices.
3. The two vertices "a" units from the center.
4. The two covertices "b" units from the center.
5. The semimajor axes = "a" units long, from the center to each vertex. 
6. The minor axis = the shortest chord that passes through the center,
bisected at the center. Its endpoints are the two covertices.
7. The semiminor axes = "b" units long, from the center to each vertex 
8. The two foci = points on the major axis "c" units from the center.
9. The formula for "c", which is c2 = a2 - b2  
10. The standard equation for an ellipse that looks like this:  which is 
11. The standard equation for an ellipse that looks like this:  which is 

We begin by plotting the given parts: the vertices and the foci:



The center is the midpoint between the two vertices, which is also
the midpoint between the two foci. By looking at the graph we see that it is
the point (4,-3).  So we plot the center (4,-3), in red below:



We see that a = 5 because we observe that it is 5 units from the center to
each vertex.

We draw the major axis, from vertex to vertex (in green). It is 10
units long, a = 5 units on each side of the center



To draw the minor axis we must calculate b, using the formula 

c2 = a2 - b2

We know that the semimajor axis is 5 units long, so a = 5. 
We know that c = 4, because we observe that it is 4 units from the center
to each focus.  We substitute these:

 42 = 52 - b2
 16 = 25 - b2
 -9 = -b2 
  9 = b2
  3 = b

Now we can draw the minor axis  (also in green) perpendicular to
the major axis bisected at the center.  It is 6 units long, b = 3
units on each side of the center.  It is the line that connects 
the two covertices:



Now its easy to sketch in the ellipse:



Now since we see that it is an ellipse that looks like this:

it has the standard equation:

 

And we substitute (h,k) = (4,-3), a = 5, and b = 3,
and the standard equation is

 

 

Edwin