The equation √[(x-2)²+(y-1)²] + √[(x+2)²+(y-4)²]=5 represents circle/ ellipse/ line segment??? Elaborate how.

The given eqn is the eqn of an ellipse.If you note carefully you will find that the eqn represents the sum of the distances of a point (x,y) from (2,1) & (-2,4) resp. which is constant & equal to 5. This basically is the definition of an ellipse. Ellipse is defined as a curve in which the sum of the distances of any point on it from the focii is constant.Here the focii of the ellipse are (2,1) & (-2,4).