If a point moves in a plane in such a way that the sum of its distances from two fixed points is constant, the curve so traced is called :

If a point moves on a plane in such a way that the sum of its distances from two fixed points on the plane is always a constant then the locus traced out by the moving point on the plane is called an ellipse and the two fixed points are the two foci of the ellipse.

to the length of the major axis of the ellipse.

Let P (x, y) be any point on the ellipse $$x^2/a^2 + y^2/b^2$$ = 1.

Let MPM' be the perpendicular through P on directrices ZK and Z'K'. Now by definition we get,

SP = e ∙ PM

⇒ SP = e ∙ NK

⇒ SP = e (CK - CN)

⇒ SP = e(ae - x)

⇒ SP = a - ex ………………..…….. (i)

and

S'P = e ∙ PM'

⇒ S'P = e ∙ (NK')

⇒ S'P = e (CK' + CN)

⇒ S'P = e (ae + x)

⇒ S'P = a + ex ………………..…….. (ii)

Therefore, SP + S'P = a - ex + a + ex = 2a = major axis.

Hence, the sum of the focal distance of a point P (x, y) on the ellipse$$ x^2/a^2 + y^2/b^2$$ = 1. is constant and equal to the length of the major axis (i.e., 2a) of the ellipse.

So , the answer would be option b)Ellipse