Question

find the vertex, focus, and directrix for the following parabolas.

(a)
\\((y-9)^2 = 16(x-4)\\)
vertex : \\((4,9)\\)
focus : \\((8,9)\\)
directrix: \\(x=0\\)

(b)
\\(y^2 - 8y = 8x - 4^2\\)
vertex : \\((4,4)\\)
focus : \\((6,4)\\)
directrix: \\(x=2\\)

Explanation:

Step1: Identify standard form parameters for (a)

$$(y-k)^2 = 4p(x-h) \implies h=4, k=9, 4p=16 \implies p=4$$

Step2: Calculate properties for (a)

$$\text{vertex: } (h,k) = (4,9), \text{ focus: } (h+p,k) = (8,9), \text{ directrix: } x = h-p = 0$$

Step3: Rewrite equation (b) in standard form

$$y^2 - 8y + 16 = 8x - 16 + 16 \implies (y-4)^2 = 8x$$

Step4: Identify standard form parameters for (b)

$$(y-k)^2 = 4p(x-h) \implies h=0, k=4, 4p=8 \implies p=2$$

Step5: Calculate properties for (b)

$$\text{vertex: } (h,k) = (0,4), \text{ focus: } (h+p,k) = (2,4), \text{ directrix: } x = h-p = -2$$

Answer:

(a) vertex: \((4,9)\), focus: \((8,9)\), directrix: \(x = 0\)
(b) vertex: \((0,4)\), focus: \((2,4)\), directrix: \(x = -2\)