# Finding equation of a parabola? Help

1. Dec 2, 2007

### dillonwerd

I know this is probably really easy, but I cant get the answer.

Use the definition of a parabola and the distance formula to find the equation of the parabola with focus (1,2) and directrix x=-1

I know the answer is x=(y-2)^2/4. Can anyone tell me how to get there? Sorry to waste your time if this should be pretty easy

2. Dec 2, 2007

### dillonwerd

I just noticed I posted this in the wrong section. Sorry about that...Go ahead and move it if necessary Mods.

3. Dec 2, 2007

### symbolipoint

Yes, it is pretty easy. You need to know a definition of a parabola: The set of points equidistant from a line and a point not on that line. If you draw this, you could approximate visually the intended parabola.

Focus: (1, 2), Directrix: x=-1

( (x -(-1))^2 + (y - y)^2)^(0.5) = ((x-1)^2 + (y-2)^2)^(0.5)

The leftside is distance from a point on parabola to the line; the rightside is distance from point on parabola
to the directrix. Translate the equation above into conventional mathematical, algebra symbolism, work the
steps, and you'll have your expected result, whatever it is.

4. Dec 3, 2007

### HallsofIvy

Staff Emeritus
In "convential mathematical, algebra symbolism" that is:
$$\sqrt{(x-(-1))^2+ (y-y)^2}= \sqrt{(x-1)^2+ (y-2)^2}$$
First obvious step is to square both sides.
The formula on the right is obviously the distance from the focus, (1, 2) to the general point (x,y) on the parabola. The formula on the left is the distance from the point (x,y) to a point on the line x= -1, (-1, y). Since the distance from a point to a vertical line is just the absolute value of the difference in the x coordinates, you could also write
$$|x-(-1)|= \sqrt{(x-1)^2+ (y-2)^2}$$
Since
$$\sqrt{(x-(-1))^2+ (y-y)^2}= \sqrt{(x-(-1))^2}= |x-(-1)|$$
Of course, you would still want to square both sides and so get rid of both the square root and the absolute value.

5. Dec 3, 2007

### symbolipoint

Dillonwerd, actually since you are trying to find steps to show the parabola expressed of
$$$x = \frac{1}{4}(y - 2)^2$$$ ,

You should change your non-typeset form to something more like:
x = (1/4)*(y - 2)^2 which would express what you really are trying to express.