Explanation:

An ellipse is a closed curve formed by the intersection of a cone and a plane. It has two main axes - the major axis and the minor axis. The major axis is the longer axis and the minor axis is the shorter axis of the ellipse.

The focal points of an ellipse are two points on the major axis, denoted as F1 and F2. The sum of the distances of any point on the ellipse from these two focal points is always constant and equal to the length of the major axis.

Proof:

Let's consider an ellipse with semi-major axis 'a' and semi-minor axis 'b'. The distance between the center of the ellipse and each focal point is denoted as 'c'.

The sum of the distances of a point on the ellipse from the two focal points is:

S = PF1 + PF2

where PF1 is the distance from the point to F1 and PF2 is the distance from the point to F2.

Using the distance formula, we can calculate PF1 and PF2 as follows:

PF1 = sqrt((x-c)^2 + y^2)
PF2 = sqrt((x+c)^2 + y^2)

where (x, y) are the coordinates of the point on the ellipse.

Substituting the values of PF1 and PF2 into the equation for S:

S = sqrt((x-c)^2 + y^2) + sqrt((x+c)^2 + y^2)

To simplify this equation, we can square both sides:

S^2 = ((x-c)^2 + y^2) + 2(sqrt((x-c)^2 + y^2) * sqrt((x+c)^2 + y^2)) + ((x+c)^2 + y^2)

Simplifying further, we get:

S^2 = 2(x^2 + c^2 + y^2) + 2(sqrt((x-c)^2 + y^2) * sqrt((x+c)^2 + y^2))

Using the equation of the ellipse, which states that (x^2/a^2) + (y^2/b^2) = 1, we can substitute (x^2 + y^2) with a^2 - (a^2/b^2) * x^2:

S^2 = 2(a^2 - (a^2/b^2) * x^2 + c^2) + 2(sqrt((x-c)^2 + y^2) * sqrt((x+c)^2 + y^2))

Simplifying further, we get:

S^2 = 2(a^2 + c^2) - (2a^2/b^2) * x^2 + 2(sqrt((x-c)^2 + y^2) * sqrt((x+c)^2 + y^2))

Notice that the term (2a^2/b^2) * x^2 is independent of the point on the ellipse. This means that it remains constant for all points on the ellipse.

Therefore, the sum of the distances of any point on the ellipse from the two focal points is constant and equal to:

S^2 = 2(a^2 + c^2)

Since the length of the major axis is 2a, we can rewrite the equation as:

S