Find the equation of a parabola with focus (2, -3) and directrix x = 5.

The equation of the parabola is (y + 3)² = -6(x - 7/2)

Step-by-step explanation:

* Lets revise the equation of a parabola

- If the equation is in the form (y − k)² = 4p(x − h), then:

• Use the given equation to identify h and k for the vertex, (h , k)

• Use the value of k to determine the axis of symmetry, y = k

• Use h , k and p to find the coordinates of the focus, (h + p , k)

• Use h and p to find the equation of the directrix, x = h − p

* Now lets solve the problem

∵ The directrix ⇒ x = 5

∴ The form is (y − k)² = 4p(x − h)

∵ The directrix is x = h - p

∴ h - p = 5 ⇒ (1)

∵ The focus is (h + p , k)

∵ The focus is (2 , -3)

∴ k = -3

∴ h + p = 2 ⇒ (2)

- Add (1) and (2) to find h

∴ 2h = 7 ⇒ ÷ 2 for both sides

∴ h = 7/2

- Substitute this value in (1) or (2) to find p

∴ 7/2 + p = 2 ⇒ subtract 7/2 from both sides

∴ p = -3/2

* Now we can write the equation

∴ (y - -3)² = 4(-3/2) (x - 7/2)

∴ (y + 3)² = -6(x - 7/2) ⇒ in standard form

* The equation of the parabola is (y + 3)² = -6(x - 7/2)