Question

(c)
\\((x - 4)^2 = 8(y - 4)\\)

vertex :
focus :
directrix :

(d)
\\(x^2 + 32x = 4y - 36\\)

vertex :
focus :
directrix :

Explanation:

Step1: Identify standard form for (c)

$$(x - h)^2 = 4p(y - k)$$

Step2: Extract vertex for (c)

$$(h, k) = (4, 4)$$

Step3: Calculate value of p for (c)

$$4p = 8 \implies p = 2$$

Step4: Find focus for (c)

$$(h, k + p) = (4, 4 + 2) = (4, 6)$$

Step5: Find directrix for (c)

$$y = k - p \implies y = 4 - 2 = 2$$

Step6: Complete square for (d)

$$x^2 + 32x + 256 = 4y - 36 + 256 \implies (x + 16)^2 = 4(y + 55)$$

Step7: Extract vertex for (d)

$$(h, k) = (-16, -55)$$

Step8: Calculate value of p for (d)

$$4p = 4 \implies p = 1$$

Step9: Find focus for (d)

$$(h, k + p) = (-16, -55 + 1) = (-16, -54)$$

Step10: Find directrix for (d)

$$y = k - p \implies y = -55 - 1 = -56$$

Answer:

(c)
vertex: \((4, 4)\)
focus: \((4, 6)\)
directrix: \(y = 2\)

(d)
vertex: \((-16, -55)\)
focus: \((-16, -54)\)
directrix: \(y = -56\)