Find the eq

Given that we need to find the equation of the parabola with vertex at the origin and the directrix(M) is y = 2.

We know that the axis is perpendicular to the directrix. Since the directrix is parallel to the x - axis, the axis will parallel to the y - axis.

Since, axis passes through the origin, the equation of the axis is x = 0.

The intersection of the point of the axis and directrix will be (0, 2).

We know that vertex is the midpoint of focus and point on directrix which lies on axis(intersection point).

Let (x₁, y₁) be the focus,

⇒

⇒

⇒ x = 0 and y + 2 = 0

⇒ x = 0 and y = - 2.

The focus is S(0, - 2).

Let P(x, y) be any point on the parabola.

We know that the point on the parabola is equidistant from focus and directrix.

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

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

⇒ SP = PM

⇒ SP² = PM²

⇒

⇒

⇒ x² + y² + 4y + 4 = y² - 4y + 4

⇒ x² = - 8y

∴The equation of the parabola is x² = - 8y.