Answer:
The equation of the parabola is (x - 4)² = 8(y - 4)
Explanation:
* Lets revise the equation of a parabola
- If the equation is in the form (x − h)² = 4p(y − k), then:
• Use the given equation to identify h and k for the vertex, (h , k)
• Use the value of h to determine the axis of symmetry, x = h
• Use h , k and p to find the coordinates of the focus, (h , k + p)
• Use k and p to find the equation of the directrix, y = k − p
* Now lets solve the problem
∵ The directrix is y = 2
∴ The equation is (x - h)² = 4p(y - k)
∴ The focus is (h , k + p)
∵ The focus is (4 , 6)
∴ h = 4
∵ k + p = 6 ⇒ (1)
∵ The directrix is y = k - p
∴ k - p = 2 ⇒ (2)
* Add (1) and(2) to find k
∴ 2k = 8 ⇒ ÷ 2 for both sides
∴ k = 4
* Substitute the value of k in (1) to find p
∵ 4 + p = 6 ⇒ subtract 4 from both sides
∴ p = 2
* Now lets write the equation
∴ (x - 4)² = 4(2)(y - 4) ⇒ simplify
∴ (x - 4)² = 8(y - 4)
* The equation of the parabola is (x - 4)² = 8(y - 4)