Question

find the focus and directrix of the following parabola:

\\(y - 2)^2 = 16(x - 7)\\

focus: \\((?, \quad )\\)
directrix: \\(x = \quad \\)

Explanation:

Identify the standard form and vertex

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

$$ (h, k) = (7, 2) $$

Calculate the focal parameter p

$$ 4p = 16 \implies p = 4 $$

Determine the focus and directrix

$$ \text{Focus: } (h + p, k) = (7 + 4, 2) = (11, 2) $$

$$ \text{Directrix: } x = h - p \implies x = 7 - 4 = 3 $$

Answer:

Find the focus and directrix of the following parabola:

$$(y - 2)^2 = 16(x - 7)$$

Focus: (<blank>11</blank>, <blank>2</blank>)
Directrix: \(x =\) <blank>3</blank>