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