Ellipse

1. Definition of an Ellipse
   An ellipse is the set of all points on a plane such that the sum of the distances from two fixed points, and , is constant. These two fixed points are called the foci of the ellipse. If a line segment joining the two foci intersects the ellipse at two points, these points are labeled as and . The perpendicular bisector of the line segment intersects the ellipse at two points, labeled and . These four points, and , are called the vertices of the ellipse.
   
   
   
   
   The line segment is called the major axis.
   The line segment is called the minor axis.
   The intersection of the major and minor axes is called the center of the ellipse.
   
   
2. Equation of an Ellipse
1. Ellipse with Horizontal Major Axis
   For an ellipse with foci at and , and the sum of the distances from the foci to any point on the ellipse is , the equation is:
   
   
2. Ellipse with Vertical Major Axis
   For an ellipse with foci at and , and the sum of the distances from the foci to any point on the ellipse is , the equation is:
   
   
3. Translation of an Ellipse
   If the ellipse is translated by units horizontally and units vertically, the new equation becomes:
4. The lengths of the major and minor axes remain unchanged after translation.