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?