Show that the equation
sqrt((x-c)^2 + y^2)+sqrt((x+c)^2 + y^2) = 2a
can be simplified to
(x^2/a^2) + (y^2/b^2) = 1
where b^2 = a^2-c^2.
Draw a diagram of an ellipse. Draw diagonals from (0,b) to (-c,0) and (c,0)
The definition of such an ellipse is that the sum of the distances from a point to the foci is constant, and = 2a where a,b,c are described above.
Just google any explanation of the definition of an ellipse, its foci, and eccentricity.