Given that we need to find the equation of set of all points whose distances from S(0,4) are of their distances from the line(M) y = 9.

Let P(x,y) be any point from the set of all points.

We know that distance between the points (x₁,y₁) and (x₂,y₂) is .

We know that the perpendicular distance from the point (x₁,y₁) to the line ax + by + c = 0 is .

⇒

⇒

⇒

⇒

⇒

⇒ 9x² + 9y² - 72y + 144 = 4y² - 72y + 324

⇒ 9x² + 5y² = 180

∴ The equation of the ellipse is 9x² + 5y² = 180.