Answer:
1) y = (1/28) x²
2) x = (1/8) y²
3) y = (-1/16) x²
4) x = (-1/12) y²
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