Derive the equation of the parabola with a focus at (6, 2) and a directrix of y = 1. f(x) = −one half(x − 6)2 + three halves f(x) = one half(x − 6)2 + three halves f(x) = −one half(x + three halves)2 + 6 f(x) = one half(x + three halves)2 + 6

A parabola is a plane that is approximately U-shaped.

The equation of the parabola is:

The given parameters are:

First, equate the directrix to 0

The equation is then calculated as:

Where:

So, we have:

Expand

Subtract y^2 from both sides

Collect like terms

Divide through by 2

Express 39 as 36 + 3

Factor out 3/2

Expand the bracket

Factorize

Factor out x - 6

Express as squares

Hence, the equation of the parabola is:

Read more about equations of parabola at:

link

Second choice.

f(x) = 1/2(x - 6)^2 + 3/2.

Step-by-step explanation:

The distance of a point  (x, y) from the focus = the distance of the point from the directrix, so:

(x - 6)^2 + (y - 2)^2 = (y - 1)^2

x^2 - 12x + 36 + y^2 - 4y + 4 = y^2 - 2y + 1

x^2 -12x + 39 = 2y

y = f(x) = 1/2 (x^2 - 12x + 39)

I see you want the answer in vertex for so it is:

f(x)  = 1/2 [ (x - 6)^2 - 36)  + 39)

f(x)  = 1/2(x - 6)^2 + 3)

f(x) = 1/2(x - 6)^2 + 3/2.

1. y = 12x2 - 9x + 4 has no real solution(s).

2. 10x + y = -x2 + 2 has two real solution(s).

3. 4y - 7 = 5x2 - x + 2 + 3y has no real solution(s).

4. y = (-x + 4)2 has one real solution(s).