Derive the equation of a parabola with a focus at (-7,5) and a directrix of y = - 11

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)