Two vertices of a triangle lie at (4, 0) and (8, 0). The perimeter of the triangle is 12 units. What are all the possible locations of the third vertex? How do you know you have found them all? Can you determine which of these vertices will produce a triangle with the largest area?
If you consider the ellipse whose foci are (4,0) and (8,0) that has major axis endpoints of (0,0) and (12,0), all the points of that ellipse (with the exception of the endpoints of the major axis) will produce a triangle with perimeter 12.
By the definition of what constitutes an ellipse, this ellipse will contan all the points.
There are two isosceles triangles with base (4,0) and (8,0). These have the largest area.