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

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)