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)

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step-by-step explanation:

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