Do I need foci to calculate an ellipse?

I have been trying to find an answer, but where I look does not tell me why I need foci if I have all 4 vertices and the center. If I am just trying to create an ellipse with the 4 vertices and center, can I just plug the numbers into the equation of an ellipse without worrying about the foci? I am trying to understand how the foci come into play, as they don't appear in the actual equation of an ellipse. However, I want my ellipse to be correct. I am trying to take a circle, and scale the y axis only, elongating the circle to create the ellipse that still passes through the 4 points, 2 now scaled. It is a vertical ellipse.

I have been trying to find an answer, but where I look does not tell me why I need foci if I have all 4 vertices and the center. If I am just trying to create an ellipse with the 4 vertices and center, can I just plug the numbers into the equation of an ellipse without worrying about the foci? I am trying to understand how the foci come into play, as they don't appear in the actual equation of an ellipse. However, I want my ellipse to be correct. I am trying to take a circle, and scale the y axis only, elongating the circle to create the ellipse that still passes through the 4 points, 2 now scaled. It is a vertical ellipse.