Focus of a parabola: y^2 = 12x

[confused0874]

I don't have a cluehow to fin the focus of a parabola. Can somebody work through my sample question, so I can take a look and figure it out.

The question is:

Find the focus of the parabola y²=12x

Thanks

 

[soroban]

Re: Focus of a parabola

Hello, confused0874!

Find the focus of the parabola $\displaystyle y^2\,=\,12x$

There are some general rules for "eyeballing" the equation . . .

$\displaystyle x^2\,=\,4py\:$ opens right or left: $\displaystyle \,\subset\,$ or $\displaystyle \,\supset$

$\displaystyle y^2\,=\,4px\;$ opems up or down: $\displaystyle \,\cup\,$ or $\displaystyle \,\cap$

If $\displaystyle 4p$ is positive, the parabola opens in the "positve" direction (right or up).
If $\displaystyle 4p$ is negative, the parabola opens in the "negative" direction (left or down).

$\displaystyle p$ is the directed distance from the vertex to the focus.
$\displaystyle \;\;$Note: the focus is always "inside the bend" of the parabola.


Your problem: $\displaystyle \,y^2\,=\,12x$

The vertex is at the origin: $\displaystyle (0,0)$

Since the $\displaystyle y$ is squared, it opens sideways (right or left).
Since the "4p" is +12, it opens to the right: $\displaystyle \,\subset$

. . . - | - - * - -
. . . - | -*- - - -
. . . - |*- - - - -
. . . - | - - - - -
. . . - * - - - o <-- focus
. . . - | - - - - -
. . . - |*- - - - -
. . . - | -*- - - -
. . . - | - - * - -

Since $\displaystyle 4p\,=\,12\;\;\Rightarrow\;\;p\,=\,3$

The focus is 3 units to the right of the vertex: $\displaystyle \,F(3,0)$

 

[confused0874]

Thanks

This site has been such a help. I have been trying to learn from a book, but the expertson here have such a good way of showing howa problemis solved.

 

[skeeter]

Re: Focus of a parabola

$\displaystyle x^2\,=\,4py\:$ opens right or left: $\displaystyle \,\subset\,$ or $\displaystyle \,\supset$

did you get these two statements(above and below) "switched" ???

$\displaystyle y^2\,=\,4px\;$ opems up or down: $\displaystyle \,\cup\,$ or $\displaystyle \,\cap$

 

[soroban]

Re: Focus of a parabola

did you get these two statements(above and below) "switched" ???

$\displaystyle y^2\,=\,4px\;$ opems up or down: $\displaystyle \,\cup\,$ or $\displaystyle \,\cap$

Yike! . . . yes, I did . . . *blush*

 

[confused0874]

Re: Focus of a parabola

did you get these two statements(above and below) "switched" ???

$\displaystyle y^2\,=\,4px\;$ opems up or down: $\displaystyle \,\cup\,$ or $\displaystyle \,\cap$

Yike! . . . yes, I did . . . *blush*

Does that change the result?