Rain equation of the parabola with vertex at origin given its directrix

1) y = (1/28) x²

2) x = (1/8) y²

3) y = (-1/16) x²

4) x = (-1/12) y²

Step-by-step explanation:

The vertex point is (0 , 0)

Directrix: y = k or x = k

The focus (a , b)

1) ∵ The directrix is y = -7 ⇒ below the vertex point by 7

   ∴ The distance between the vertex and the focus = 7

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (0 , 7)

By using the rule of the distance

∵ (y - -7)² = (x - 0)² + (y - 7)²

∴ (y + 7)² = x² + (y - 7)²

∴ y² + 14y + 49 = x² + y² - 14y + 49

∴ 28y = x²

∴ y = (1/28) x²

2) ∵ The directrix is x = -2 ⇒ left the vertex point by 2

   ∴ The distance between the vertex and the focus = 2

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (2 , 0)

By using the rule of the distance

∵ (x - -2)² = (x - 2)² + (y - 0)²

∴ (x + 2)² = (x - 2)² + y²

∴ x² + 4x + 4 = x² - 4x + 4 + y²

∴ 8x = y²

∴ x = (1/8) y²

3) ∵ The directrix is y = 4 ⇒ over the vertex point by 4

   ∴ The distance between the vertex and the focus = 4

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (0 , -4)

By using the rule of the distance

∵ (y - 4)² = (x - 0)² + (y - -4)²

∴ (y - 4)² = x² + (y + 4)²

∴ y² - 8y + 16 = x² + y² + 8y + 16

∴ -16y = x²

∴ y = (-1/16) x²

4) ∵ The directrix is x = 3 ⇒ right to the vertex point by 3

   ∴ The distance between the vertex and the focus = 3

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (-3 , 0)

By using the rule of the distance

∵ (x - 3)² = (x - -3)² + (y - 0)²

∴ (x - 3)² = (x + 3)² + y²

∴ x² - 6x + 9 = x² + 6x + 9 + y²

∴ -12x = y²

∴ x = (-1/12) y²

step-by-step explanation:

i have done this all on a piece of paper but idk how to put it on here cuz it needs to be drawn

i'm assuming that it's c.)   volcanoes built the mountains and earthquakes tore them down.