Question

for the given model of the form $y^{2}=4 p x$,
(a) determine the value of $p$.
(b) identify the focus of the parabola.
(c) write an equation for the directrix.
express numbers in exact, simplest form.
$y^{2}=40 x$

Explanation:

Step1: Determine the value of \( p \)

Given the equation \( y^{2}=4px \) and \( y^{2} = 40x \).
By comparing the two equations \( 4p=40 \), then \( p=\frac{40}{4}=10 \).

Step2: Identify the focus of the parabola

For a parabola of the form \( y^{2}=4px \), the focus is at the point \((p,0)\).
Since \( p = 10 \), the focus is \((10,0)\).

Step3: Write an equation for the directrix

For a parabola of the form \( y^{2}=4px \), the equation of the directrix is \( x=-p \).
Since \( p = 10 \), the directrix is \( x = - 10 \).

Answer:

(a) \( p = 10 \)
(b) Focus: \((10,0)\)
(c) Directrix: \( x=-10 \)