Answer:
The equation of the parabola is (x + 7)² = 32(y + 3)
Step-by-step explanation:
* Lets revise the equation of the parabola in standard form
- The standard form is (x - h)² = 4p(y - k)
- The focus is (h , k + p)
- The directrix is y = k - p
* Lets solve the problem
- The parabola has focus at (-7 , 5) and a directrix of y = -11
∵ The focus is (h , k + p)
∵ The focus at (-7 , 5)
∴ h = -7
∴ k + p = 5 ⇒ (1)
∵ The directrix is y = k - p
∵ The directrix of y = -11
∴ k - p = -11 ⇒ (2)
- Add equation (1) and (2) to find k and p
∴ 2k = -6
- Divide both sides by 2
∴ k = -3
- substitute the value of k in equation (1)
∴ -3 + p = 5
- Add 3 to both sides
∴ p = 8
∵ The form of the equation of the parabola is (x - h)² = 4p(y - k)
∴ (x - -7)² = 4(8)(y - -3)
# Remember ⇒ (-)(-) = (+)
∴ (x + 7)² = 32(y + 3)
* The equation of the parabola is (x + 7)² = 32(y + 3)
