Finding formula of parabola

Find a formula of the parabola with emphasis at factor $(0,5)$ whose directrix is the line $y=0$. (Derive this formula making use of the definition of the parabola as a set of factors that are equidistant from the directrix and also the emphasis)

Ok this set is eliminating me. My book has this

A formula of the parabola with emphasis $(0,p)$ and also directrix $y=-p$ is $x^2=4py$

However, my trouble has actually $(0,p)$ with p being 5, yet the directrix is not $p=-y=-5$, it is $0$. Therefore I attempted acquiring it by hand as they claimed by

$\sqrt{x^2+(y-p)^2}=|y+p|$

But that simply takes me back to the initial formula I stated, $x^2=4py$. Not exactly sure what I'm intended to do.

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2019-12-02 02:51:10
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Answers: 3

From wikipedia :

Equation for a basic parabola with an emphasis factor $F =(u, v)$, and also a directrix in the kind

$$n_1x+n_2y+c=0 \,$$ is

$$\frac{\left|n_1x+n_2y+c\right|}{\sqrt{{n_1}^{2}+{n_2}^{2}}}=\sqrt{\left(x-u\right)^2+\left(y-v\right)^2} $$

So make the replacements and also validate that the what you get concurs with Arturo Magidin's derivation.

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2019-12-03 04:17:36
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So, you have a method for creating a formula for a parabola where the emphasis goes to $(0,p)$ and also the directrix is $y=-p$, today you intend to write a formula for a parabola with emphasis $(0,5)$ and also directrix $y=0$, which does not fit the formula.

If you take the problems you are offered and also convert them down 2.5 devices, the emphasis would certainly go to $(0,2.5)$ and also the directrix would certainly go to $y=-2.5$, so you need to have the ability to write a formula for that parabola. Currently, convert that formula up 2.5 devices (change $y$ with $(y-2.5)$).

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2019-12-03 04:16:47
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Edited to attempt to make clear OP is complication :

The range in between $(x,y)$ and also $(0,5)$ is without a doubt $\sqrt{x^2+(y-5)^2}$.

The range in between $(x,y)$ and also the line $y=0$ is $|y|$ (if $y>0$, after that it is over the $X$ - axis, and also the range is simply the $y$ coordinate ; if $y<0$, after that the range is $-y = |y|$). So the factors on the parabola are specifically the factors for which both ranges are equivalent, that is, all $(x,y)$ for which : $$\sqrt{x^2+(y-5)^2} = |y|.$$ Now square both sides, and also you'll get the formula of the parabola you desire.


Just how is this pertaining to the formula you price estimate? It is actually all all the same procedure. If the directrix is either straight (a line of the kind $y=k$) or upright (a line of the kind $x=\ell$), after that it is really simple to calculate the range from an indicate the directrix : the factor $(x,y)$ is $|y-k|$ far from the line $y=k$, and also is $|x-\ell|$ far from the line $x=\ell$. If the emphasis of the parabola goes to $(a,b)$, after that you desire the range from $(x,y)$ to $(a,b)$ to amount to the range to the directrix, so you get : \begin{align*} \sqrt{(x-a)^2 + (y-b)^2} &= |y-k| &\qquad&\mbox{if the directrix is $y=k$,}\\ \sqrt{(x-a)^2 + (y-b)^2} &= |x-\ell| &&\mbox{if the directrix is $x=\ell$.} \end{align*} Then making even both sides generates the formula of the parabola. Doing it appropriately will negate the $y^2$ term when the directrix is straight, and also the $x^2$ term when the directrix is upright.

For even more basic parabolas, when the directrix is neither straight neither upright yet an approximate line $y=mx+b$, you require to function a little bit harder, due to the fact that the range from $y=mx+b$ to $(x,y)$ is not so straightforward to calculate. Not also tough, yet not as straightforward. As soon as you have a formula for the range, you set it equivalent to the range to the emphasis, square both sides, and also get the formula of the parabola.

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2019-12-03 04:12:55
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