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

a - base

b - exponent

so,